# scipy.spatial.transform.Rotation¶

class scipy.spatial.transform.Rotation

Rotation in 3 dimensions.

This class provides an interface to initialize from and represent rotations with:

• Quaternions

• Rotation Matrices

• Rotation Vectors

• Modified Rodrigues Parameters

• Euler Angles

The following operations on rotations are supported:

• Application on vectors

• Rotation Composition

• Rotation Inversion

• Rotation Indexing

Indexing within a rotation is supported since multiple rotation transforms can be stored within a single Rotation instance.

To create Rotation objects use from_... methods (see examples below). Rotation(...) is not supposed to be instantiated directly.

Notes

Examples

>>> from scipy.spatial.transform import Rotation as R


A Rotation instance can be initialized in any of the above formats and converted to any of the others. The underlying object is independent of the representation used for initialization.

Consider a counter-clockwise rotation of 90 degrees about the z-axis. This corresponds to the following quaternion (in scalar-last format):

>>> r = R.from_quat([0, 0, np.sin(np.pi/4), np.cos(np.pi/4)])


The rotation can be expressed in any of the other formats:

>>> r.as_matrix()
array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
[ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
>>> r.as_rotvec()
array([0.        , 0.        , 1.57079633])
>>> r.as_euler('zyx', degrees=True)
array([90.,  0.,  0.])


The same rotation can be initialized using a rotation matrix:

>>> r = R.from_matrix([[0, -1, 0],
...                    [1, 0, 0],
...                    [0, 0, 1]])


Representation in other formats:

>>> r.as_quat()
array([0.        , 0.        , 0.70710678, 0.70710678])
>>> r.as_rotvec()
array([0.        , 0.        , 1.57079633])
>>> r.as_euler('zyx', degrees=True)
array([90.,  0.,  0.])


The rotation vector corresponding to this rotation is given by:

>>> r = R.from_rotvec(np.pi/2 * np.array([0, 0, 1]))


Representation in other formats:

>>> r.as_quat()
array([0.        , 0.        , 0.70710678, 0.70710678])
>>> r.as_matrix()
array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
[ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
>>> r.as_euler('zyx', degrees=True)
array([90.,  0.,  0.])


The from_euler method is quite flexible in the range of input formats it supports. Here we initialize a single rotation about a single axis:

>>> r = R.from_euler('z', 90, degrees=True)


Again, the object is representation independent and can be converted to any other format:

>>> r.as_quat()
array([0.        , 0.        , 0.70710678, 0.70710678])
>>> r.as_matrix()
array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
[ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
>>> r.as_rotvec()
array([0.        , 0.        , 1.57079633])


It is also possible to initialize multiple rotations in a single instance using any of the from_… functions. Here we initialize a stack of 3 rotations using the from_euler method:

>>> r = R.from_euler('zyx', [
... [90, 0, 0],
... [0, 45, 0],
... [45, 60, 30]], degrees=True)


The other representations also now return a stack of 3 rotations. For example:

>>> r.as_quat()
array([[0.        , 0.        , 0.70710678, 0.70710678],
[0.        , 0.38268343, 0.        , 0.92387953],
[0.39190384, 0.36042341, 0.43967974, 0.72331741]])


Applying the above rotations onto a vector:

>>> v = [1, 2, 3]
>>> r.apply(v)
array([[-2.        ,  1.        ,  3.        ],
[ 2.82842712,  2.        ,  1.41421356],
[ 2.24452282,  0.78093109,  2.89002836]])


A Rotation instance can be indexed and sliced as if it were a single 1D array or list:

>>> r.as_quat()
array([[0.        , 0.        , 0.70710678, 0.70710678],
[0.        , 0.38268343, 0.        , 0.92387953],
[0.39190384, 0.36042341, 0.43967974, 0.72331741]])
>>> p = r[0]
>>> p.as_matrix()
array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
[ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])
>>> q = r[1:3]
>>> q.as_quat()
array([[0.        , 0.38268343, 0.        , 0.92387953],
[0.39190384, 0.36042341, 0.43967974, 0.72331741]])


In fact it can be converted to numpy.array:

>>> r_array = np.asarray(r)
>>> r_array.shape
(3,)
>>> r_array[0].as_matrix()
array([[ 2.22044605e-16, -1.00000000e+00,  0.00000000e+00],
[ 1.00000000e+00,  2.22044605e-16,  0.00000000e+00],
[ 0.00000000e+00,  0.00000000e+00,  1.00000000e+00]])


Multiple rotations can be composed using the * operator:

>>> r1 = R.from_euler('z', 90, degrees=True)
>>> r2 = R.from_rotvec([np.pi/4, 0, 0])
>>> v = [1, 2, 3]
>>> r2.apply(r1.apply(v))
array([-2.        , -1.41421356,  2.82842712])
>>> r3 = r2 * r1 # Note the order
>>> r3.apply(v)
array([-2.        , -1.41421356,  2.82842712])


Finally, it is also possible to invert rotations:

>>> r1 = R.from_euler('z', [90, 45], degrees=True)
>>> r2 = r1.inv()
>>> r2.as_euler('zyx', degrees=True)
array([[-90.,   0.,   0.],
[-45.,   0.,   0.]])


These examples serve as an overview into the Rotation class and highlight major functionalities. For more thorough examples of the range of input and output formats supported, consult the individual method’s examples.

Attributes
single

Whether this instance represents a single rotation.

Methods

 Number of rotations contained in this object. Initialize from quaternions. Initialize from rotation matrix. Initialize from rotation vectors. Initialize from Modified Rodrigues Parameters (MRPs). Initialize from Euler angles. Represent as quaternions. Represent as rotation matrix. Represent as rotation vectors. Represent as Modified Rodrigues Parameters (MRPs). Represent as Euler angles. Apply this rotation to a set of vectors. Compose this rotation with the other. Invert this rotation. Get the magnitude(s) of the rotation(s). Get the mean of the rotations. Reduce this rotation with the provided rotation groups. Create a 3D rotation group. Extract rotation(s) at given index(es) from object. Get identity rotation(s). Generate uniformly distributed rotations. Estimate a rotation to optimally align two sets of vectors.