Class FieldRotation<T extends CalculusFieldElement<T>>
 Type Parameters:
T
 the type of the field elements
 All Implemented Interfaces:
Serializable
Rotation
using CalculusFieldElement
.
Instance of this class are guaranteed to be immutable.
 See Also:

Constructor Summary
ConstructorDescriptionFieldRotation
(Field<T> field, Rotation r) Build aFieldRotation
from aRotation
.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. 
Method Summary
Modifier and TypeMethodDescriptionvoid
applyInverseTo
(double[] in, T[] out) Apply the inverse of the rotation to a vector stored in an array.Apply the inverse of the instance to another rotation.Apply the inverse of the rotation to a vector.Apply the inverse of the instance to another rotation.static <T extends CalculusFieldElement<T>>
FieldRotation<T>applyInverseTo
(Rotation rOuter, FieldRotation<T> rInner) Apply the inverse of a rotation to another rotation.static <T extends CalculusFieldElement<T>>
FieldVector3D<T>applyInverseTo
(Rotation r, FieldVector3D<T> u) Apply the inverse of a rotation to a vector.Apply the inverse of the rotation to a vector.void
applyInverseTo
(T[] in, T[] out) Apply the inverse of the rotation to a vector stored in an array.void
Apply the rotation to a vector stored in an array.applyTo
(FieldRotation<T> r) Apply the instance to another rotation.applyTo
(FieldVector3D<T> u) Apply the rotation to a vector.Apply the instance to another rotation.static <T extends CalculusFieldElement<T>>
FieldRotation<T>applyTo
(Rotation r1, FieldRotation<T> rInner) Apply a rotation to another rotation.static <T extends CalculusFieldElement<T>>
FieldVector3D<T>applyTo
(Rotation r, FieldVector3D<T> u) Apply a rotation to a vector.Apply the rotation to a vector.void
Apply the rotation to a vector stored in an array.compose
(FieldRotation<T> r, RotationConvention convention) Compose the instance with another rotation.compose
(Rotation r, RotationConvention convention) Compose the instance with another rotation.composeInverse
(FieldRotation<T> r, RotationConvention convention) Compose the inverse of the instance with another rotation.composeInverse
(Rotation r, RotationConvention convention) Compose the inverse of the instance with another rotation.static <T extends CalculusFieldElement<T>>
Tdistance
(FieldRotation<T> r1, FieldRotation<T> r2) Compute the distance between two rotations.getAngle()
Get the angle of the rotation.T[]
getAngles
(RotationOrder order, RotationConvention convention) Get the Cardan or Euler angles corresponding to the instance.getAxis
(RotationConvention convention) Get the normalized axis of the rotation.static <T extends CalculusFieldElement<T>>
FieldRotation<T>getIdentity
(Field<T> field) Get identity rotation.T[][]
Get the 3X3 matrix corresponding to the instancegetQ0()
Get the scalar coordinate of the quaternion.getQ1()
Get the first coordinate of the vectorial part of the quaternion.getQ2()
Get the second coordinate of the vectorial part of the quaternion.getQ3()
Get the third coordinate of the vectorial part of the quaternion.revert()
Revert a rotation.Convert to a constant vector without derivatives.

Constructor Details

FieldRotation
Build a rotation from the quaternion coordinates.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.
 Parameters:
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 them

FieldRotation
public FieldRotation(FieldVector3D<T> axis, T angle, RotationConvention convention) throws MathIllegalArgumentException Build a rotation from an axis and an angle.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.
 Parameters:
axis
 axis around which to rotateangle
 rotation angle.convention
 convention to use for the semantics of the angle Throws:
MathIllegalArgumentException
 if the axis norm is zero

FieldRotation
Build aFieldRotation
from aRotation
. Parameters:
field
 field for the componentsr
 rotation to convert

FieldRotation
Build a rotation from a 3X3 matrix.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.
 Parameters:
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) Throws:
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 negative

FieldRotation
public FieldRotation(FieldVector3D<T> u1, FieldVector3D<T> u2, FieldVector3D<T> v1, FieldVector3D<T> v2) throws MathRuntimeException 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 (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.
 Parameters:
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 Throws:
MathRuntimeException
 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)

FieldRotation
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.
 Parameters:
u
 origin vectorv
 desired image of u by the rotation Throws:
MathRuntimeException
 if the norm of one of the vectors is zero

FieldRotation
public FieldRotation(RotationOrder order, RotationConvention convention, T alpha1, T alpha2, T 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).
 Parameters:
order
 order of rotations to compose, from left to right (i.e. we will user1.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 rotation


Method Details

getIdentity
Get identity rotation. Type Parameters:
T
 the type of the field elements Parameters:
field
 field for the components Returns:
 a new rotation

revert
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

getQ0
Get the scalar coordinate of the quaternion. Returns:
 scalar coordinate of the quaternion

getQ1
Get the first coordinate of the vectorial part of the quaternion. Returns:
 first coordinate of the vectorial part of the quaternion

getQ2
Get the second coordinate of the vectorial part of the quaternion. Returns:
 second coordinate of the vectorial part of the quaternion

getQ3
Get the third coordinate of the vectorial part of the quaternion. Returns:
 third coordinate of the vectorial part of the quaternion

getAxis
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. Parameters:
convention
 convention to use for the semantics of the angle Returns:
 normalized axis of the rotation
 See Also:

getAngle
Get the angle of the rotation. Returns:
 angle of the rotation (between 0 and π)
 See Also:

getAngles
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 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:
 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!
 Parameters:
order
 rotation order to useconvention
 convention to use for the semantics of the angle Returns:
 an array of three angles, in the order specified by the set

getMatrix
Get the 3X3 matrix corresponding to the instance Returns:
 the matrix corresponding to the instance

toRotation
Convert to a constant vector without derivatives. Returns:
 a constant vector

applyTo
Apply the rotation to a vector. Parameters:
u
 vector to apply the rotation to Returns:
 a new vector which is the image of u by the rotation

applyTo
Apply the rotation to a vector. Parameters:
u
 vector to apply the rotation to Returns:
 a new vector which is the image of u by the rotation

applyTo
Apply the rotation to a vector stored in an array. Parameters:
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)

applyTo
Apply the rotation to a vector stored in an array. Parameters:
in
 an array with three items which stores vector to rotateout
 an array with three items to put result to

applyTo
public static <T extends CalculusFieldElement<T>> FieldVector3D<T> applyTo(Rotation r, FieldVector3D<T> u) Apply a rotation to a vector. Type Parameters:
T
 the type of the field elements Parameters:
r
 rotation to applyu
 vector to apply the rotation to Returns:
 a new vector which is the image of u by the rotation

applyInverseTo
Apply the inverse of the rotation to a vector. Parameters:
u
 vector to apply the inverse of the rotation to Returns:
 a new vector which such that u is its image by the rotation

applyInverseTo
Apply the inverse of the rotation to a vector. Parameters:
u
 vector to apply the inverse of the rotation to Returns:
 a new vector which such that u is its image by the rotation

applyInverseTo
Apply the inverse of the rotation to a vector stored in an array. Parameters:
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)

applyInverseTo
Apply the inverse of the rotation to a vector stored in an array. Parameters:
in
 an array with three items which stores vector to rotateout
 an array with three items to put result to

applyInverseTo
public static <T extends CalculusFieldElement<T>> FieldVector3D<T> applyInverseTo(Rotation r, FieldVector3D<T> u) Apply the inverse of a rotation to a vector. Type Parameters:
T
 the type of the field elements Parameters:
r
 rotation to applyu
 vector to apply the inverse of the rotation to Returns:
 a new vector which such that u is its image by the rotation

applyTo
Apply the instance to another rotation.Calling this method is equivalent to call
compose(r, RotationConvention.VECTOR_OPERATOR)
. Parameters:
r
 rotation to apply the rotation to Returns:
 a new rotation which is the composition of r by the instance

compose
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 : letu
be any vector andv
its image byr1
(i.e.r1.applyTo(u) = v
). Letw
be the image ofv
by rotationr2
(i.e.r2.applyTo(v) = w
). Thenw = comp.applyTo(u)
, wherecomp = 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 allr1
,r2
,u
,v
,w
andcomp
as above,comp
could also be computed ascomp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)
. Parameters:
r
 rotation to apply the rotation toconvention
 convention to use for the semantics of the angle Returns:
 a new rotation which is the composition of r by the instance

applyTo
Apply the instance to another rotation.Calling this method is equivalent to call
compose(r, RotationConvention.VECTOR_OPERATOR)
. Parameters:
r
 rotation to apply the rotation to Returns:
 a new rotation which is the composition of r by the instance

compose
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 : letu
be any vector andv
its image byr1
(i.e.r1.applyTo(u) = v
). Letw
be the image ofv
by rotationr2
(i.e.r2.applyTo(v) = w
). Thenw = comp.applyTo(u)
, wherecomp = 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 allr1
,r2
,u
,v
,w
andcomp
as above,comp
could also be computed ascomp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)
. Parameters:
r
 rotation to apply the rotation toconvention
 convention to use for the semantics of the angle Returns:
 a new rotation which is the composition of r by the instance

applyTo
public static <T extends CalculusFieldElement<T>> FieldRotation<T> applyTo(Rotation r1, FieldRotation<T> rInner) Apply a rotation to another rotation. Applying a rotation to another rotation is computing the composition in an order compliant with the following rule : let u be any vector and v its image by rInner (i.e. rInner.applyTo(u) = v), let w be the image of v by rOuter (i.e. rOuter.applyTo(v) = w), then w = comp.applyTo(u), where comp = applyTo(rOuter, rInner). Type Parameters:
T
 the type of the field elements Parameters:
r1
 rotation to applyrInner
 rotation to apply the rotation to Returns:
 a new rotation which is the composition of r by the instance

applyInverseTo
Apply the inverse of the instance to another rotation.Calling this method is equivalent to call
composeInverse(r, RotationConvention.VECTOR_OPERATOR)
. Parameters:
r
 rotation to apply the rotation to Returns:
 a new rotation which is the composition of r by the inverse of the instance

composeInverse
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 : letu
be any vector andv
its image byr1
(i.e.r1.applyTo(u) = v
). Letw
be the inverse image ofv
byr2
(i.e.r2.applyInverseTo(v) = w
). Thenw = comp.applyTo(u)
, wherecomp = 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 allr1
,r2
,u
,v
,w
andcomp
as above,comp
could also be computed ascomp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)
. Parameters:
r
 rotation to apply the rotation toconvention
 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

applyInverseTo
Apply the inverse of the instance to another rotation.Calling this method is equivalent to call
composeInverse(r, RotationConvention.VECTOR_OPERATOR)
. Parameters:
r
 rotation to apply the rotation to Returns:
 a new rotation which is the composition of r by the inverse of the instance

composeInverse
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 : letu
be any vector andv
its image byr1
(i.e.r1.applyTo(u) = v
). Letw
be the inverse image ofv
byr2
(i.e.r2.applyInverseTo(v) = w
). Thenw = comp.applyTo(u)
, wherecomp = 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 allr1
,r2
,u
,v
,w
andcomp
as above,comp
could also be computed ascomp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)
. Parameters:
r
 rotation to apply the rotation toconvention
 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

applyInverseTo
public static <T extends CalculusFieldElement<T>> FieldRotation<T> applyInverseTo(Rotation rOuter, FieldRotation<T> rInner) Apply the inverse of a rotation to another rotation. Applying the inverse of a rotation to another rotation is computing the composition in an order compliant with the following rule : let u be any vector and v its image by rInner (i.e. rInner.applyTo(u) = v), let w be the inverse image of v by rOuter (i.e. rOuter.applyInverseTo(v) = w), then w = comp.applyTo(u), where comp = applyInverseTo(rOuter, rInner). Type Parameters:
T
 the type of the field elements Parameters:
rOuter
 rotation to apply the rotation torInner
 rotation to apply the rotation to Returns:
 a new rotation which is the composition of r by the inverse of the instance

distance
public static <T extends CalculusFieldElement<T>> T distance(FieldRotation<T> r1, FieldRotation<T> r2) 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: \(r_1(r) = r_2\)
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).
 Type Parameters:
T
 the type of the field elements Parameters:
r1
 first rotationr2
 second rotation Returns:
 distance between r1 and r2
