# `quaternions`¶

Functions to operate on, or return, quaternions

The module also includes functions for the closely related angle, axis pair as a specification for rotation.

Quaternions here consist of 4 values `w, x, y, z`, where `w` is the real (scalar) part, and `x, y, z` are the complex (vector) part.

Note - rotation matrices here apply to column vectors, that is, they are applied on the left of the vector. For example:

```>>> import numpy as np
>>> from nibabel.quaternions import quat2mat
>>> q = [0, 1, 0, 0] # 180 degree rotation around axis 0
>>> M = quat2mat(q) # from this module
>>> vec = np.array([1, 2, 3]).reshape((3,1)) # column vector
>>> tvec = np.dot(M, vec)
```
 `angle_axis2mat`(theta, vector[, is_normalized]) Rotation matrix of angle theta around vector `angle_axis2quat`(theta, vector[, is_normalized]) Quaternion for rotation of angle theta around vector Conjugate of quaternion Return identity quaternion `fillpositive`(xyz[, w2_thresh]) Compute unit quaternion from last 3 values Return multiplicative inverse of quaternion q Return True is this is very nearly a unit quaternion Calculate quaternion corresponding to given rotation matrix `mult`(q1, q2) Multiply two quaternions `nearly_equivalent`(q1, q2[, rtol, atol]) Returns True if q1 and q2 give near equivalent transforms Return norm of quaternion `quat2angle_axis`(quat[, identity_thresh]) Convert quaternion to rotation of angle around axis Calculate rotation matrix corresponding to quaternion `rotate_vector`(v, q) Apply transformation in quaternion q to vector v

## angle_axis2mat¶

nibabel.quaternions.angle_axis2mat(theta, vector, is_normalized=False)

Rotation matrix of angle theta around vector

Parameters:
thetascalar

angle of rotation

vector3 element sequence

vector specifying axis for rotation.

is_normalizedbool, optional

True if vector is already normalized (has norm of 1). Default False

Returns:
matarray shape (3,3)

rotation matrix for specified rotation

Notes

## angle_axis2quat¶

nibabel.quaternions.angle_axis2quat(theta, vector, is_normalized=False)

Quaternion for rotation of angle theta around vector

Parameters:
thetascalar

angle of rotation

vector3 element sequence

vector specifying axis for rotation.

is_normalizedbool, optional

True if vector is already normalized (has norm of 1). Default False

Returns:
quat4 element sequence of symbols

quaternion giving specified rotation

Notes

Examples

```>>> q = angle_axis2quat(np.pi, [1, 0, 0])
>>> np.allclose(q, [0, 1, 0,  0])
True
```

## conjugate¶

nibabel.quaternions.conjugate(q)

Conjugate of quaternion

Parameters:
q4 element sequence

w, i, j, k of quaternion

Returns:
conjqarray shape (4,)

w, i, j, k of conjugate of q

## eye¶

nibabel.quaternions.eye()

Return identity quaternion

## fillpositive¶

nibabel.quaternions.fillpositive(xyz, w2_thresh=None)

Compute unit quaternion from last 3 values

Parameters:
xyziterable

iterable containing 3 values, corresponding to quaternion x, y, z

w2_threshNone or float, optional

threshold to determine if w squared is non-zero. If None (default) then w2_thresh set equal to 3 * `np.finfo(xyz.dtype).eps`, if possible, otherwise 3 * `np.finfo(np.float64).eps`

Returns:
wxyzarray shape (4,)

Full 4 values of quaternion

Notes

If w, x, y, z are the values in the full quaternion, assumes w is positive.

Gives error if w*w is estimated to be negative

w = 0 corresponds to a 180 degree rotation

The unit quaternion specifies that np.dot(wxyz, wxyz) == 1.

If w is positive (assumed here), w is given by:

w = np.sqrt(1.0-(x*x+y*y+z*z))

w2 = 1.0-(x*x+y*y+z*z) can be near zero, which will lead to numerical instability in sqrt. Here we use the system maximum float type to reduce numerical instability

Examples

```>>> import numpy as np
>>> wxyz = fillpositive([0,0,0])
>>> np.all(wxyz == [1, 0, 0, 0])
True
>>> wxyz = fillpositive([1,0,0]) # Corner case; w is 0
>>> np.all(wxyz == [0, 1, 0, 0])
True
>>> np.dot(wxyz, wxyz)
1.0
```

## inverse¶

nibabel.quaternions.inverse(q)

Return multiplicative inverse of quaternion q

Parameters:
q4 element sequence

w, i, j, k of quaternion

Returns:
invqarray shape (4,)

w, i, j, k of quaternion inverse

## isunit¶

nibabel.quaternions.isunit(q)

Return True is this is very nearly a unit quaternion

## mat2quat¶

nibabel.quaternions.mat2quat(M)

Calculate quaternion corresponding to given rotation matrix

Parameters:
Marray-like

3x3 rotation matrix

Returns:
q(4,) array

closest quaternion to input matrix, having positive q[0]

Notes

Method claimed to be robust to numerical errors in M

Constructs quaternion by calculating maximum eigenvector for matrix K (constructed from input M). Although this is not tested, a maximum eigenvalue of 1 corresponds to a valid rotation.

A quaternion q*-1 corresponds to the same rotation as q; thus the sign of the reconstructed quaternion is arbitrary, and we return quaternions with positive w (q[0]).

References

Examples

```>>> import numpy as np
>>> q = mat2quat(np.eye(3)) # Identity rotation
>>> np.allclose(q, [1, 0, 0, 0])
True
>>> q = mat2quat(np.diag([1, -1, -1]))
>>> np.allclose(q, [0, 1, 0, 0]) # 180 degree rotn around axis 0
True
```

## mult¶

nibabel.quaternions.mult(q1, q2)

Multiply two quaternions

Parameters:
q14 element sequence
q24 element sequence
Returns:
q12shape (4,) array

Notes

## nearly_equivalent¶

nibabel.quaternions.nearly_equivalent(q1, q2, rtol=1e-05, atol=1e-08)

Returns True if q1 and q2 give near equivalent transforms

q1 may be nearly numerically equal to q2, or nearly equal to q2 * -1 (because a quaternion multiplied by -1 gives the same transform).

Parameters:
q14 element sequence

w, x, y, z of first quaternion

q24 element sequence

w, x, y, z of second quaternion

Returns:
equivbool

True if q1 and q2 are nearly equivalent, False otherwise

Examples

```>>> q1 = [1, 0, 0, 0]
>>> nearly_equivalent(q1, [0, 1, 0, 0])
False
>>> nearly_equivalent(q1, [1, 0, 0, 0])
True
>>> nearly_equivalent(q1, [-1, 0, 0, 0])
True
```

## norm¶

nibabel.quaternions.norm(q)

Return norm of quaternion

Parameters:
q4 element sequence

w, i, j, k of quaternion

Returns:
nscalar

quaternion norm

## quat2angle_axis¶

nibabel.quaternions.quat2angle_axis(quat, identity_thresh=None)

Convert quaternion to rotation of angle around axis

Parameters:
quat4 element sequence

w, x, y, z forming quaternion

identity_threshNone or scalar, optional

threshold below which the norm of the vector part of the quaternion (x, y, z) is deemed to be 0, leading to the identity rotation. None (the default) leads to a threshold estimated based on the precision of the input.

Returns:
thetascalar

angle of rotation

vectorarray shape (3,)

axis around which rotation occurs

Notes

A quaternion for which x, y, z are all equal to 0, is an identity rotation. In this case we return a 0 angle and an arbitrary vector, here [1, 0, 0]

Examples

```>>> theta, vec = quat2angle_axis([0, 1, 0, 0])
>>> np.allclose(theta, np.pi)
True
>>> vec
array([1., 0., 0.])
```

If this is an identity rotation, we return a zero angle and an arbitrary vector

```>>> quat2angle_axis([1, 0, 0, 0])
(0.0, array([1., 0., 0.]))
```

## quat2mat¶

nibabel.quaternions.quat2mat(q)

Calculate rotation matrix corresponding to quaternion

Parameters:
q4 element array-like
Returns:
M(3,3) array

Rotation matrix corresponding to input quaternion q

Notes

Rotation matrix applies to column vectors, and is applied to the left of coordinate vectors. The algorithm here allows non-unit quaternions.

References

Algorithm from https://en.wikipedia.org/wiki/Rotation_matrix#Quaternion

Examples

```>>> import numpy as np
>>> M = quat2mat([1, 0, 0, 0]) # Identity quaternion
>>> np.allclose(M, np.eye(3))
True
>>> M = quat2mat([0, 1, 0, 0]) # 180 degree rotn around axis 0
>>> np.allclose(M, np.diag([1, -1, -1]))
True
```

## rotate_vector¶

nibabel.quaternions.rotate_vector(v, q)

Apply transformation in quaternion q to vector v

Parameters:
v3 element sequence

3 dimensional vector

q4 element sequence

w, i, j, k of quaternion

Returns:
vdasharray shape (3,)

v rotated by quaternion q

Notes