public class Rotation
extends java.lang.Object
implements java.io.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 implicitely 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 followong 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 rotation transforms the coordinates coordinates of this vector in inertial frame into the coordinates of the same vector in satellite frame. In this case, the rotation implicitely 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 directionf at rest in a topocentric frame into the sighting direction in the same topocentric frame. 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.
These examples show that a rotation is what the user wants it to be, so 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.
Vector3D
,
RotationOrder
,
Serialized FormConstructor and Description 

Rotation()
Build the identity rotation.

Rotation(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,
double alpha1,
double alpha2,
double alpha3)
Build a rotation from three Cardan or Euler elementary rotations.

Rotation(Vector3D axis,
double angle)
Build a rotation from an axis and an angle.

Rotation(Vector3D u,
Vector3D v)
Build one of the rotations that transform one vector into another one.

Rotation(Vector3D u1,
Vector3D u2,
Vector3D v1,
Vector3D v2)
Build the rotation that transforms a pair of vector into another pair.

Modifier and Type  Method and Description 

Rotation 
applyInverseTo(Rotation r)
Apply the inverse of the instance to another rotation.

Vector3D 
applyInverseTo(Vector3D u)
Apply the inverse of the rotation to a vector.

Rotation 
applyTo(Rotation r)
Apply the instance to another rotation.

Vector3D 
applyTo(Vector3D u)
Apply the rotation to a vector.

double 
getAngle()
Get the angle of the rotation.

double[] 
getAngles(RotationOrder order)
Get the Cardan or Euler angles corresponding to the instance.

Vector3D 
getAxis()
Get the normalized axis of the rotation.

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

double 
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.

Rotation 
revert()
Revert a rotation.

public Rotation()
public Rotation(double q0, double q1, double q2, double 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.
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 Rotation(Vector3D axis, double 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 PI/2 as the
angle to this constructor, and then apply
the
instance to +i, we will get +j.
axis
 axis around which to rotateangle
 rotation angle.java.lang.ArithmeticException
 if the axis norm is zeropublic Rotation(double[][] m, double threshold) throws NotARotationMatrixException
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)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 negativepublic Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2)
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}) 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 rotationjava.lang.IllegalArgumentException
 if the norm of one of the vectors is zeropublic Rotation(Vector3D u, Vector3D v)
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 rotationjava.lang.IllegalArgumentException
 if the norm of one of the vectors is zeropublic Rotation(RotationOrder order, double alpha1, double alpha2, double alpha3)
Cardan rotations are three successive rotations around the canonical axes X, Y and Z, each axis beeing 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 beeing 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 rotationpublic Rotation revert()
public double getQ0()
public double getQ1()
public double getQ2()
public double getQ3()
public Vector3D getAxis()
public double getAngle()
public double[] getAngles(RotationOrder order) throws CardanEulerSingularityException
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 useCardanEulerSingularityException
 if the rotation is singular with respect to the angles set
specifiedpublic double[][] getMatrix()
public Vector3D applyTo(Vector3D u)
u
 vector to apply the rotation topublic Vector3D applyInverseTo(Vector3D u)
u
 vector to apply the inverse of the rotation topublic Rotation applyTo(Rotation r)
r
 rotation to apply the rotation topublic Rotation applyInverseTo(Rotation r)
r
 rotation to apply the rotation toprocessing library peasycam by Jonathan Feinberg. (c) 2013