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.
<Comment:
|value: '.. versionadded: 1.2.0'
|>
Rotation in 3 dimensions.
>>> 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 :None:None:`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.
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.spatial.transform._rotation_spline.RotationSpline
scipy.spatial.transform._rotation.Slerp
scipy.spatial.transform._rotation.Rotation
Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.
Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)
SVG is more flexible but power hungry; and does not scale well to 50 + nodes.
All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them