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.

See also

Slerp

Notes

New in version 1.2.0.

Examples

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

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

__len__

Number of rotations contained in this object.

from_quat(type cls, quat)

Initialize from quaternions.

from_matrix(type cls, matrix)

Initialize from rotation matrix.

from_rotvec(type cls, rotvec[, degrees])

Initialize from rotation vectors.

from_mrp(type cls, mrp)

Initialize from Modified Rodrigues Parameters (MRPs).

from_euler(type cls, seq, angles[, degrees])

Initialize from Euler angles.

as_quat(self)

Represent as quaternions.

as_matrix(self)

Represent as rotation matrix.

as_rotvec(self[, degrees])

Represent as rotation vectors.

as_mrp(self)

Represent as Modified Rodrigues Parameters (MRPs).

as_euler(self, seq[, degrees])

Represent as Euler angles.

concatenate(type cls, rotations)

Concatenate a sequence of Rotation objects.

apply(self, vectors[, inverse])

Apply this rotation to a set of vectors.

__mul__

Compose this rotation with the other.

inv(self)

Invert this rotation.

magnitude(self)

Get the magnitude(s) of the rotation(s).

mean(self[, weights])

Get the mean of the rotations.

reduce(self[, left, right, return_indices])

Reduce this rotation with the provided rotation groups.

create_group(type cls, group[, axis])

Create a 3D rotation group.

__getitem__

Extract rotation(s) at given index(es) from object.

identity(type cls[, num])

Get identity rotation(s).

random(type cls[, num, random_state])

Generate uniformly distributed rotations.

align_vectors(type cls, a, b[, weights, ...])

Estimate a rotation to optimally align two sets of vectors.