Class Rotation
 All Implemented Interfaces:
Serializable
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
useroriented 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, 1.0e10).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 = r_{1} o
r_{2}
(which means that for each vector u
,
r(u) = r_{1}(r_{2}(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 r_{1}
to r_{2}
and the result
we get is r = r_{1} o r_{2}
. For this purpose, the
class provides the methods: applyTo(Rotation)
and
applyInverseTo(Rotation)
.
Rotations are guaranteed to be immutable objects.
 See Also:

Field Summary

Constructor Summary
ConstructorDescriptionRotation
(double[][] m, double threshold) Build a rotation from a 3X3 matrix.Rotation
(double q0, double q1, double q2, double q3, boolean needsNormalization) Build a rotation from the quaternion coordinates.Rotation
(RotationOrder order, RotationConvention convention, double alpha1, double alpha2, double alpha3) Build a rotation from three Cardan or Euler elementary rotations.Rotation
(Vector3D axis, double angle, RotationConvention convention) Build a rotation from an axis and an angle.Build one of the rotations that transform one vector into another one.Build the rotation that transforms a pair of vectors into another pair. 
Method Summary
Modifier and TypeMethodDescriptionvoid
applyInverseTo
(double[] in, double[] 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.void
applyTo
(double[] in, double[] out) Apply the rotation to a vector stored in an array.Apply the instance to another rotation.Apply the rotation to a vector.compose
(Rotation r, RotationConvention convention) Compose the instance with another rotation.composeInverse
(Rotation r, RotationConvention convention) Compose the inverse of the instance with another rotation.static double
Compute the distance between two rotations.double
getAngle()
Get the angle of the rotation.double[]
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.double[][]
Get the 3X3 matrix corresponding to the instancedouble
getQ0()
Get the scalar coordinate of the quaternion.double
getQ1()
Get the first coordinate of the vectorial part of the quaternion.double
getQ2()
Get the second coordinate of the vectorial part of the quaternion.double
getQ3()
Get the third coordinate of the vectorial part of the quaternion.revert()
Revert a rotation.

Field Details

IDENTITY
Identity rotation.


Constructor Details

Rotation
public Rotation(double q0, double q1, double q2, double q3, boolean needsNormalization) 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

Rotation
public Rotation(Vector3D axis, double angle, RotationConvention convention) throws MathIllegalArgumentException Build a rotation from an axis and an angle. Parameters:
axis
 axis around which to rotateangle
 rotation angleconvention
 convention to use for the semantics of the angle Throws:
MathIllegalArgumentException
 if the axis norm is zero

Rotation
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

Rotation
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)

Rotation
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

Rotation
public Rotation(RotationOrder order, RotationConvention convention, double alpha1, double alpha2, double 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

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
public double getQ0()Get the scalar coordinate of the quaternion. Returns:
 scalar coordinate of the quaternion

getQ1
public double getQ1()Get the first coordinate of the vectorial part of the quaternion. Returns:
 first coordinate of the vectorial part of the quaternion

getQ2
public double getQ2()Get the second coordinate of the vectorial part of the quaternion. Returns:
 second coordinate of the vectorial part of the quaternion

getQ3
public double 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
public double 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
public double[][] getMatrix()Get the 3X3 matrix corresponding to the instance Returns:
 the matrix corresponding to the instance

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
public void applyTo(double[] in, double[] out) 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)

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
public void applyInverseTo(double[] in, double[] out) 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)

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

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

distance
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).
 Parameters:
r1
 first rotationr2
 second rotation Returns:
 distance between r1 and r2
