## Class Rotation

• All Implemented Interfaces:
`Serializable`

```public class Rotation
extends Object
implements Serializable```
This class implements rotations in a three-dimensional space.

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 user-oriented 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, 1.0e-10).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 = r1 o r2``` (which means that for each vector `u`, `r(u) = r1(r2(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 `r1` to `r2` and the result we get is `r = r1 o r2`. For this purpose, the class provides the methods: `applyTo(Rotation)` and `applyInverseTo(Rotation)`.

Rotations are guaranteed to be immutable objects.

`Vector3D`, `RotationOrder`, Serialized Form
• ### Field Summary

Fields
Modifier and Type Field Description
`static Rotation` `IDENTITY`
Identity rotation.
• ### Constructor Summary

Constructors
Constructor Description
```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)```
```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)```
```Rotation​(Vector3D axis, double angle, RotationConvention convention)```
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 vectors into another pair.
• ### Method Summary

All Methods
Modifier and Type Method Description
`void` ```applyInverseTo​(double[] in, double[] out)```
Apply the inverse of the rotation to a vector stored in an array.
`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.
`void` ```applyTo​(double[] in, double[] out)```
Apply the rotation to a vector stored in an array.
`Rotation` `applyTo​(Rotation r)`
Apply the instance to another rotation.
`Vector3D` `applyTo​(Vector3D u)`
Apply the rotation to a vector.
`Rotation` ```compose​(Rotation r, RotationConvention convention)```
Compose the instance with another rotation.
`Rotation` ```composeInverse​(Rotation r, RotationConvention convention)```
Compose the inverse of the instance with another rotation.
`static double` ```distance​(Rotation r1, Rotation r2)```
Compute the distance between two rotations.
`double` `getAngle()`
Get the angle of the rotation.
`double[]` `getAngles​(RotationOrder order)`
`double[]` ```getAngles​(RotationOrder order, RotationConvention convention)```
Get the Cardan or Euler angles corresponding to the instance.
`Vector3D` `getAxis()`
Deprecated.
`Vector3D` `getAxis​(RotationConvention convention)`
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.
• ### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ### Field Detail

• #### IDENTITY

`public static final Rotation IDENTITY`
Identity rotation.
• ### Constructor Detail

• #### 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 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.

Parameters:
`q0` - scalar part of the quaternion
`q1` - first coordinate of the vectorial part of the quaternion
`q2` - second coordinate of the vectorial part of the quaternion
`q3` - third coordinate of the vectorial part of the quaternion
`needsNormalization` - 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 rotate
`angle` - rotation angle
`convention` - convention to use for the semantics of the angle
Throws:
`MathIllegalArgumentException` - if the axis norm is zero
• #### Rotation

```public Rotation​(double[][] m,
double threshold)
throws MathIllegalArgumentException```
Build a rotation from a 3X3 matrix.

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.

Parameters:
`m` - rotation matrix
`threshold` - 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

```public Rotation​(Vector3D u1,
Vector3D u2,
Vector3D v1,
Vector3D 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 (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) half-plane.

Parameters:
`u1` - first vector of the origin pair
`u2` - second vector of the origin pair
`v1` - desired image of u1 by the rotation
`v2` - 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

```public Rotation​(Vector3D u,
Vector3D v)
throws MathRuntimeException```
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 vector
`v` - 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 use `r1.compose(r2.compose(r3, convention), convention)`)
`convention` - convention to use for the semantics of the angle
`alpha1` - angle of the first elementary rotation
`alpha2` - angle of the second elementary rotation
`alpha3` - angle of the third elementary rotation
• ### Method Detail

• #### revert

`public 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

`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

`public Vector3D getAxis​(RotationConvention convention)`
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
`Rotation(Vector3D, double, RotationConvention)`
• #### getAngle

`public double getAngle()`
Get the angle of the rotation.
Returns:
angle of the rotation (between 0 and π)
`Rotation(Vector3D, double)`
• #### getAngles

```@Deprecated
public double[] getAngles​(RotationOrder order)
throws MathIllegalStateException```
Get the Cardan or Euler angles corresponding to the instance.

Calling this method is equivalent to call `getAngles(order, RotationConvention.VECTOR_OPERATOR)`

Parameters:
`order` - rotation order to use
Returns:
an array of three angles, in the order specified by the set
Throws:
`MathIllegalStateException` - if the rotation is singular with respect to the angles set specified
• #### getAngles

```public double[] getAngles​(RotationOrder order,
RotationConvention convention)
throws MathIllegalStateException```
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 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:

• 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 use
`convention` - convention to use for the semantics of the angle
Returns:
an array of three angles, in the order specified by the set
Throws:
`MathIllegalStateException` - if the rotation is singular with respect to the angles set specified
• #### getMatrix

`public double[][] getMatrix()`
Get the 3X3 matrix corresponding to the instance
Returns:
the matrix corresponding to the instance
• #### applyTo

`public Vector3D applyTo​(Vector3D u)`
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 rotate
`out` - an array with three items to put result to (it can be the same array as in)
• #### applyInverseTo

`public Vector3D applyInverseTo​(Vector3D u)`
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 rotate
`out` - an array with three items to put result to (it can be the same array as in)
• #### applyTo

`public Rotation applyTo​(Rotation r)`
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

```public Rotation compose​(Rotation r,
RotationConvention convention)```
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 : 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)`.

Parameters:
`r` - rotation to apply the rotation to
`convention` - convention to use for the semantics of the angle
Returns:
a new rotation which is the composition of r by the instance
• #### applyInverseTo

`public Rotation applyInverseTo​(Rotation r)`
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

```public Rotation composeInverse​(Rotation r,
RotationConvention convention)```
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 : 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)`.

Parameters:
`r` - rotation to apply the rotation to
`convention` - 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

```public static double distance​(Rotation r1,
Rotation 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:

```        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).

Parameters:
`r1` - first rotation
`r2` - second rotation
Returns:
distance between r1 and r2