T
 the type of the field elementspublic class FieldRotation<T extends CalculusFieldElement<T>> extends Object implements Serializable
Rotation
using CalculusFieldElement
.
Instance of this class are guaranteed to be immutable.
FieldVector3D
,
RotationOrder
,
Serialized FormConstructor and Description 

FieldRotation(Field<T> field,
Rotation r)
Build a
FieldRotation from a Rotation . 
FieldRotation(FieldVector3D<T> u,
FieldVector3D<T> v)
Build one of the rotations that transform one vector into another one.

FieldRotation(FieldVector3D<T> u1,
FieldVector3D<T> u2,
FieldVector3D<T> v1,
FieldVector3D<T> v2)
Build the rotation that transforms a pair of vectors into another pair.

FieldRotation(FieldVector3D<T> axis,
T angle,
RotationConvention convention)
Build a rotation from an axis and an angle.

FieldRotation(RotationOrder order,
RotationConvention convention,
T alpha1,
T alpha2,
T alpha3)
Build a rotation from three Cardan or Euler elementary rotations.

FieldRotation(T[][] m,
double threshold)
Build a rotation from a 3X3 matrix.

FieldRotation(T q0,
T q1,
T q2,
T q3,
boolean needsNormalization)
Build a rotation from the quaternion coordinates.

Modifier and Type  Method and Description 

void 
applyInverseTo(double[] in,
T[] out)
Apply the inverse of the rotation to a vector stored in an array.

FieldRotation<T> 
applyInverseTo(FieldRotation<T> r)
Apply the inverse of the instance to another rotation.

FieldVector3D<T> 
applyInverseTo(FieldVector3D<T> u)
Apply the inverse of the rotation to a vector.

FieldRotation<T> 
applyInverseTo(Rotation r)
Apply the inverse of the instance to another rotation.

static <T extends CalculusFieldElement<T>> 
applyInverseTo(Rotation rOuter,
FieldRotation<T> rInner)
Apply the inverse of a rotation to another rotation.

static <T extends CalculusFieldElement<T>> 
applyInverseTo(Rotation r,
FieldVector3D<T> u)
Apply the inverse of a rotation to a vector.

void 
applyInverseTo(T[] in,
T[] out)
Apply the inverse of the rotation to a vector stored in an array.

FieldVector3D<T> 
applyInverseTo(Vector3D u)
Apply the inverse of the rotation to a vector.

void 
applyTo(double[] in,
T[] out)
Apply the rotation to a vector stored in an array.

FieldRotation<T> 
applyTo(FieldRotation<T> r)
Apply the instance to another rotation.

FieldVector3D<T> 
applyTo(FieldVector3D<T> u)
Apply the rotation to a vector.

FieldRotation<T> 
applyTo(Rotation r)
Apply the instance to another rotation.

static <T extends CalculusFieldElement<T>> 
applyTo(Rotation r1,
FieldRotation<T> rInner)
Apply a rotation to another rotation.

static <T extends CalculusFieldElement<T>> 
applyTo(Rotation r,
FieldVector3D<T> u)
Apply a rotation to a vector.

void 
applyTo(T[] in,
T[] out)
Apply the rotation to a vector stored in an array.

FieldVector3D<T> 
applyTo(Vector3D u)
Apply the rotation to a vector.

FieldRotation<T> 
compose(FieldRotation<T> r,
RotationConvention convention)
Compose the instance with another rotation.

FieldRotation<T> 
compose(Rotation r,
RotationConvention convention)
Compose the instance with another rotation.

FieldRotation<T> 
composeInverse(FieldRotation<T> r,
RotationConvention convention)
Compose the inverse of the instance with another rotation.

FieldRotation<T> 
composeInverse(Rotation r,
RotationConvention convention)
Compose the inverse of the instance with another rotation.

static <T extends CalculusFieldElement<T>> 
distance(FieldRotation<T> r1,
FieldRotation<T> r2)
Compute the distance between two rotations.

T 
getAngle()
Get the angle of the rotation.

T[] 
getAngles(RotationOrder order,
RotationConvention convention)
Get the Cardan or Euler angles corresponding to the instance.

FieldVector3D<T> 
getAxis(RotationConvention convention)
Get the normalized axis of the rotation.

static <T extends CalculusFieldElement<T>> 
getIdentity(Field<T> field)
Get identity rotation.

T[][] 
getMatrix()
Get the 3X3 matrix corresponding to the instance

T 
getQ0()
Get the scalar coordinate of the quaternion.

T 
getQ1()
Get the first coordinate of the vectorial part of the quaternion.

T 
getQ2()
Get the second coordinate of the vectorial part of the quaternion.

T 
getQ3()
Get the third coordinate of the vectorial part of the quaternion.

FieldRotation<T> 
revert()
Revert a rotation.

Rotation 
toRotation()
Convert to a constant vector without derivatives.

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 q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2} = 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 4^{th} component and the vector part as the first three components. This is not our convention. We put the scalar part as the first component.
q0
 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 quaternionneedsNormalization
 if true, the coordinates are considered
not to be normalized, a normalization preprocessing step is performed
before using thempublic FieldRotation(FieldVector3D<T> axis, T angle, RotationConvention convention) 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) }. Note the minus sign on the angle!
On the one hand this convention is consistent with a vectorial perspective (moving vectors in fixed frames), on the other hand it is different from conventions with a frame perspective (fixed vectors viewed from different frames) like the ones used for example in spacecraft attitude community or in the graphics community.
axis
 axis around which to rotateangle
 rotation angle.convention
 convention to use for the semantics of the angleMathIllegalArgumentException
 if the axis norm is zeropublic FieldRotation(Field<T> field, Rotation r)
FieldRotation
from a Rotation
.field
 field for the componentsr
 rotation to convertpublic FieldRotation(T[][] m, double threshold) throws MathIllegalArgumentException
Rotation matrices are orthogonal matrices, i.e. unit matrices (which are matrices for which m.m^{T} = 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)MathIllegalArgumentException
 if the matrix is not a 3X3
matrix, or if it cannot be transformed into an orthogonal matrix
with the given threshold, or if the determinant of the resulting
orthogonal matrix is negativepublic FieldRotation(FieldVector3D<T> u1, FieldVector3D<T> u2, FieldVector3D<T> v1, FieldVector3D<T> v2) throws MathRuntimeException
Except for possible scale factors, if the instance were applied to the pair (u_{1}, u_{2}) it will produce the pair (v_{1}, v_{2}).
If the angular separation between u_{1} and u_{2} is not the same as the angular separation between v_{1} and v_{2}, then a corrected v'_{2} will be used rather than v_{2}, the corrected vector will be in the (±v_{1}, +v_{2}) halfplane.
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 rotationMathRuntimeException
 if the norm of one of the vectors is zero,
or if one of the pair is degenerated (i.e. the vectors of the pair are collinear)public FieldRotation(FieldVector3D<T> u, FieldVector3D<T> v) throws MathRuntimeException
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.
u
 origin vectorv
 desired image of u by the rotationMathRuntimeException
 if the norm of one of the vectors is zeropublic FieldRotation(RotationOrder order, RotationConvention convention, 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 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 anglealpha1
 angle of the first elementary rotationalpha2
 angle of the second elementary rotationalpha3
 angle of the third elementary rotationpublic static <T extends CalculusFieldElement<T>> FieldRotation<T> getIdentity(Field<T> field)
T
 the type of the field elementsfield
 field for the componentspublic FieldRotation<T> revert()
public T getQ0()
public T getQ1()
public T getQ2()
public T getQ3()
public FieldVector3D<T> getAxis(RotationConvention convention)
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.
convention
 convention to use for the semantics of the angleFieldRotation(FieldVector3D, CalculusFieldElement, RotationConvention)
public T getAngle()
FieldRotation(FieldVector3D, CalculusFieldElement, RotationConvention)
public T[] getAngles(RotationOrder order, RotationConvention convention) throws MathIllegalStateException
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_{1}, a_{2} and a_{3} is the same as the rotation defined by the angles π + a_{1}, π  a_{2} and π + a_{3}. This method implements the following arbitrary choices:
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!
order
 rotation order to useconvention
 convention to use for the semantics of the angleMathIllegalStateException
 if the rotation is
singular with respect to the angles set specifiedpublic T[][] getMatrix()
public Rotation toRotation()
public FieldVector3D<T> applyTo(FieldVector3D<T> u)
u
 vector to apply the rotation topublic FieldVector3D<T> applyTo(Vector3D u)
u
 vector to apply the rotation topublic 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 topublic static <T extends CalculusFieldElement<T>> FieldVector3D<T> applyTo(Rotation r, FieldVector3D<T> u)
T
 the type of the field elementsr
 rotation to applyu
 vector to apply the rotation topublic FieldVector3D<T> applyInverseTo(FieldVector3D<T> u)
u
 vector to apply the inverse of the rotation topublic FieldVector3D<T> applyInverseTo(Vector3D u)
u
 vector to apply the inverse of the rotation topublic 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 topublic static <T extends CalculusFieldElement<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 topublic FieldRotation<T> applyTo(FieldRotation<T> r)
Calling this method is equivalent to call
compose(r, RotationConvention.VECTOR_OPERATOR)
.
r
 rotation to apply the rotation topublic FieldRotation<T> compose(FieldRotation<T> r, RotationConvention convention)
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)
.
r
 rotation to apply the rotation toconvention
 convention to use for the semantics of the anglepublic FieldRotation<T> applyTo(Rotation r)
Calling this method is equivalent to call
compose(r, RotationConvention.VECTOR_OPERATOR)
.
r
 rotation to apply the rotation topublic FieldRotation<T> compose(Rotation r, RotationConvention convention)
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)
.
r
 rotation to apply the rotation toconvention
 convention to use for the semantics of the anglepublic static <T extends CalculusFieldElement<T>> FieldRotation<T> applyTo(Rotation r1, FieldRotation<T> rInner)
T
 the type of the field elementsr1
 rotation to applyrInner
 rotation to apply the rotation topublic FieldRotation<T> applyInverseTo(FieldRotation<T> r)
Calling this method is equivalent to call
composeInverse(r, RotationConvention.VECTOR_OPERATOR)
.
r
 rotation to apply the rotation topublic FieldRotation<T> composeInverse(FieldRotation<T> r, RotationConvention convention)
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)
.
r
 rotation to apply the rotation toconvention
 convention to use for the semantics of the anglepublic FieldRotation<T> applyInverseTo(Rotation r)
Calling this method is equivalent to call
composeInverse(r, RotationConvention.VECTOR_OPERATOR)
.
r
 rotation to apply the rotation topublic FieldRotation<T> composeInverse(Rotation r, RotationConvention convention)
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)
.
r
 rotation to apply the rotation toconvention
 convention to use for the semantics of the anglepublic static <T extends CalculusFieldElement<T>> FieldRotation<T> applyInverseTo(Rotation rOuter, FieldRotation<T> rInner)
T
 the type of the field elementsrOuter
 rotation to apply the rotation torInner
 rotation to apply the rotation topublic static <T extends CalculusFieldElement<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:
r_{1}(r) = r_{2}
This distance is an angle between 0 and π. Its value is the smallest possible upper bound of the angle in radians between r_{1}(v) and r_{2}(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 rotationCopyright © 20162022 CS GROUP. All rights reserved.