org.apache.commons.math3.geometry.euclidean.threed.FieldRotation<T>
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java.lang.Object
T - the type of the field elementspublic class FieldRotation<T extends RealFieldElement<T>>
This class is a re-implementation of Rotation using RealFieldElement.
Instance of this class are guaranteed to be immutable.
FieldVector3D,
RotationOrder,
Serialized Form| Constructor Summary | |
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FieldRotation(FieldVector3D<T> u,
FieldVector3D<T> v)
Build one of the rotations that transform one vector into another one. |
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FieldRotation(FieldVector3D<T> u1,
FieldVector3D<T> u2,
FieldVector3D<T> v1,
FieldVector3D<T> v2)
Build the rotation that transforms a pair of vector into another pair. |
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FieldRotation(FieldVector3D<T> axis,
T angle)
Build a rotation from an axis and an angle. |
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FieldRotation(RotationOrder order,
T alpha1,
T alpha2,
T alpha3)
Build a rotation from three Cardan or Euler elementary rotations. |
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FieldRotation(T[][] m,
double threshold)
Build a rotation from a 3X3 matrix. |
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FieldRotation(T q0,
T q1,
T q2,
T q3,
boolean needsNormalization)
Build a rotation from the quaternion coordinates. |
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| Method Summary | ||
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void |
applyInverseTo(double[] in,
T[] out)
Apply the inverse of the rotation to a vector stored in an array. |
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FieldRotation<T> |
applyInverseTo(FieldRotation<T> r)
Apply the inverse of the instance to another rotation. |
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FieldVector3D<T> |
applyInverseTo(FieldVector3D<T> u)
Apply the inverse of the rotation to a vector. |
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FieldRotation<T> |
applyInverseTo(Rotation r)
Apply the inverse of the instance to another rotation. |
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static
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applyInverseTo(Rotation rOuter,
FieldRotation<T> rInner)
Apply the inverse of the a rotation to another rotation. |
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static
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applyInverseTo(Rotation r,
FieldVector3D<T> u)
Apply the inverse of a rotation to a vector. |
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void |
applyInverseTo(T[] in,
T[] out)
Apply the inverse of the rotation to a vector stored in an array. |
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FieldVector3D<T> |
applyInverseTo(Vector3D u)
Apply the inverse of the rotation to a vector. |
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void |
applyTo(double[] in,
T[] out)
Apply the rotation to a vector stored in an array. |
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FieldRotation<T> |
applyTo(FieldRotation<T> r)
Apply the instance to another rotation. |
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FieldVector3D<T> |
applyTo(FieldVector3D<T> u)
Apply the rotation to a vector. |
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FieldRotation<T> |
applyTo(Rotation r)
Apply the instance to another rotation. |
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static
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applyTo(Rotation r1,
FieldRotation<T> rInner)
Apply a rotation to another rotation. |
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static
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applyTo(Rotation r,
FieldVector3D<T> u)
Apply a rotation to a vector. |
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void |
applyTo(T[] in,
T[] out)
Apply the rotation to a vector stored in an array. |
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FieldVector3D<T> |
applyTo(Vector3D u)
Apply the rotation to a vector. |
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static
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distance(FieldRotation<T> r1,
FieldRotation<T> r2)
Compute the distance between two rotations. |
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T |
getAngle()
Get the angle of the rotation. |
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T[] |
getAngles(RotationOrder order)
Get the Cardan or Euler angles corresponding to the instance. |
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FieldVector3D<T> |
getAxis()
Get the normalized axis of the rotation. |
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T[][] |
getMatrix()
Get the 3X3 matrix corresponding to the instance |
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T |
getQ0()
Get the scalar coordinate of the quaternion. |
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T |
getQ1()
Get the first coordinate of the vectorial part of the quaternion. |
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T |
getQ2()
Get the second coordinate of the vectorial part of the quaternion. |
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T |
getQ3()
Get the third coordinate of the vectorial part of the quaternion. |
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FieldRotation<T> |
revert()
Revert a rotation. |
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Rotation |
toRotation()
Convert to a constant vector without derivatives. |
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| Methods inherited from class java.lang.Object |
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clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait |
| Constructor Detail |
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public FieldRotation(T q0,
T q1,
T q2,
T q3,
boolean needsNormalization)
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. As a consequence, the normalized quaternion coordinates are defined as :
q0 = cos(θ/2)
q1 = x * sin(θ/2)
q2 = y * sin(θ/2)
q3 = z * sin(θ/2) }
θ beeing the oriented rotation angle around the axis (x, y, z).
needsNormalization - if true, the coordinates are considered
not to be normalized, a normalization preprocessing step is performed
before using themq0 - scalar part of the quaternionq1 - first coordinate of the vectorial part of the quaternionq2 - second coordinate of the vectorial part of the quaternionq3 - third coordinate of the vectorial part of the quaternion
MathIllegalArgumentException - thrown if norm of the quaternion is zero
public FieldRotation(FieldVector3D<T> axis,
T angle)
throws MathIllegalArgumentException
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
apply the instance to +i, we will get
+j.
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) }. No minus sign on the angle!
axis - axis around which to rotateangle - rotation angle.
MathIllegalArgumentException - if the axis norm is zero
public FieldRotation(T[][] m,
double threshold)
throws NotARotationMatrixException
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.
m - rotation matrixthreshold - 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)
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(FieldVector3D<T> u1,
FieldVector3D<T> u2,
FieldVector3D<T> v1,
FieldVector3D<T> v2)
throws MathArithmeticException
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) plane.
u1 - first vector of the origin pairu2 - second vector of the origin pairv1 - desired image of u1 by the rotationv2 - desired image of u2 by the rotation
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 colinear)
public FieldRotation(FieldVector3D<T> u,
FieldVector3D<T> v)
throws MathArithmeticException
Except for a possible scale factor, if the instance were applied to the vector u it will produce the vector v. There is an infinite number of such rotations, this constructor choose the one with the smallest associated angle (i.e. the one whose axis is orthogonal to the (u, v) plane). If u and v are colinear, an arbitrary rotation axis is chosen.
u - origin vectorv - desired image of u by the rotation
MathArithmeticException - if the norm of one of the vectors is zero
public FieldRotation(RotationOrder order,
T alpha1,
T alpha2,
T alpha3)
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).
order - order of rotations to usealpha1 - angle of the first elementary rotationalpha2 - angle of the second elementary rotationalpha3 - angle of the third elementary rotation| Method Detail |
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public FieldRotation<T> revert()
public T getQ0()
public T getQ1()
public T getQ2()
public T getQ3()
public FieldVector3D<T> getAxis()
FieldRotation(FieldVector3D, RealFieldElement)public T getAngle()
FieldRotation(FieldVector3D, RealFieldElement)public T[] getAngles(RotationOrder order)
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:
Cardan and Euler angle have a very disappointing drawback: all of them have singularities. 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!
order - rotation order to use
public T[][] getMatrix()
public Rotation toRotation()
public FieldVector3D<T> applyTo(FieldVector3D<T> u)
u - vector to apply the rotation to
public FieldVector3D<T> applyTo(Vector3D u)
u - vector to apply the rotation to
public void applyTo(T[] in,
T[] out)
in - an array with three items which stores vector to rotateout - an array with three items to put result to (it can be the same
array as in)
public void applyTo(double[] in,
T[] out)
in - an array with three items which stores vector to rotateout - an array with three items to put result to
public static <T extends RealFieldElement<T>> FieldVector3D<T> applyTo(Rotation r,
FieldVector3D<T> u)
T - the type of the field elementsr - rotation to applyu - vector to apply the rotation to
public FieldVector3D<T> applyInverseTo(FieldVector3D<T> u)
u - vector to apply the inverse of the rotation to
public FieldVector3D<T> applyInverseTo(Vector3D u)
u - vector to apply the inverse of the rotation to
public void applyInverseTo(T[] in,
T[] out)
in - an array with three items which stores vector to rotateout - an array with three items to put result to (it can be the same
array as in)
public void applyInverseTo(double[] in,
T[] out)
in - an array with three items which stores vector to rotateout - an array with three items to put result to
public static <T extends RealFieldElement<T>> FieldVector3D<T> applyInverseTo(Rotation r,
FieldVector3D<T> u)
T - the type of the field elementsr - rotation to applyu - vector to apply the inverse of the rotation to
public FieldRotation<T> applyTo(FieldRotation<T> r)
Applying the instance to a rotation is creating a composed rotation in the following order :
R3 = R2.applyTo(R1) is equivalent to R3 = R2 o R1
Example :
The vector u being transformed into v by R1 : v = R1(u)
The vector v being transformed into w by R2 : w = R2(v)
w = R2(v) = R2(R1(u)) = R2 o R1(u) = R3(u)
r - rotation to apply the rotation to
public FieldRotation<T> applyTo(Rotation r)
Applying the instance to a rotation is creating a composed rotation in the following order :
R3 = R2.applyTo(R1) is equivalent to R3 = R2 o R1
Example :
The vector u being transformed into v by R1 : v = R1(u)
The vector v being transformed into w by R2 : w = R2(v)
w = R2(v) = R2(R1(u)) = R2 o R1(u) = R3(u)
r - rotation to apply the rotation to
public static <T extends RealFieldElement<T>> FieldRotation<T> applyTo(Rotation r1,
FieldRotation<T> rInner)
Applying the instance to a rotation is creating a composed rotation in the following order :
R3 = R2.applyTo(R1) is equivalent to R3 = R2 o R1
Example :
The vector u being transformed into v by R1 : v = R1(u)
The vector v being transformed into w by R2 : w = R2(v)
w = R2(v) = R2(R1(u)) = R2 o R1(u) = R3(u)
T - the type of the field elementsr1 - rotation to applyrInner - rotation to apply the rotation to
public FieldRotation<T> applyInverseTo(FieldRotation<T> r)
Applying the inverse of the instance to a rotation is creating a composed rotation in the following order :
R3 = R2.applyInverseTo(R1) is equivalent to R3 = R2-1 o R1
Example :
The vector u being transformed into v by R1 : v = R1(u)
The vector v being transformed into w by R2-1 : w = R2-1(v)
w = R2-1(v) = R2-1(R1(u)) = R2-1 o R1(u) = R3(u)
r - rotation to apply the rotation to
public FieldRotation<T> applyInverseTo(Rotation r)
Applying the inverse of the instance to a rotation is creating a composed rotation in the following order :
R3 = R2.applyInverseTo(R1) is equivalent to R3 = R2-1 o R1
Example :
The vector u being transformed into v by R1 : v = R1(u)
The vector v being transformed into w by R2-1 : w = R2-1(v)
w = R2-1(v) = R2-1(R1(u)) = R2-1 o R1(u) = R3(u)
r - rotation to apply the rotation to
public static <T extends RealFieldElement<T>> FieldRotation<T> applyInverseTo(Rotation rOuter,
FieldRotation<T> rInner)
Applying the inverse of the instance to a rotation is creating a composed rotation in the following order :
R3 = R2.applyInverseTo(R1) is equivalent to R3 = R2-1 o R1
Example :
The vector u being transformed into v by R1 : v = R1(u)
The vector v being transformed into w by R2-1 : w = R2-1(v)
w = R2-1(v) = R2-1(R1(u)) = R2-1 o R1(u) = R3(u)
T - the type of the field elementsrOuter - rotation to apply the rotation torInner - rotation to apply the rotation to
public static <T extends RealFieldElement<T>> T distance(FieldRotation<T> r1,
FieldRotation<T> r2)
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).
T - the type of the field elementsr1 - first rotationr2 - second rotation
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