# quaternion

Source: quaternion/__init__.py

Adds a quaternion dtype to NumPy.

## Class quaternion

Floating-point quaternion numbers

## allclose

allclose(
a, b, rtol=8.881784197001252e-16, atol=0.0, equal_nan=False, verbose=False
)


Source: quaternion/__init__.py

Returns True if two arrays are element-wise equal within a tolerance. This function is essentially a wrapper for the quaternion.isclose function, but returns a single boolean value of True if all elements of the output from quaternion.isclose are True, and False otherwise. This function also adds the option. Note that this function has stricter tolerances than the numpy.allclose function, as well as the additional verbose option.

### Parameters

• a, b: array_like

Input arrays to compare.

• rtol: float

The relative tolerance parameter (see Notes).

• atol: float

The absolute tolerance parameter (see Notes).

• equal_nan: bool

Whether to compare NaN's as equal. If True, NaN's in a will be considered equal to NaN's in b in the output array.

• verbose: bool

If the return value is False, all the non-close values are printed, iterating through the non-close indices in order, displaying the array values along with the index, with a separate line for each pair of values.

### Returns

• allclose: bool

Returns True if the two arrays are equal within the given tolerance; False otherwise.

!!! note If the following equation is element-wise True, then allclose returns True. absolute(a - b) <= (atol + rtol * absolute(b)) The above equation is not symmetric in a and b, so that allclose(a, b) might be different from allclose(b, a) in some rare cases.

## as_euler_angles

as_euler_angles(q)


Source: quaternion/__init__.py

Open Pandora's Box If somebody is trying to make you use Euler angles, tell them no, and walk away, and go and tell your mum. You don't want to use Euler angles. They are awful. Stay away. It's one thing to convert from Euler angles to quaternions; at least you're moving in the right direction. But to go the other way?! It's just not right. Assumes the Euler angles correspond to the quaternion R via R = exp(alphaz/2) * exp(betay/2) * exp(gammaz/2) The angles are naturally in radians. NOTE: Before opening an issue reporting something "wrong" with this function, be sure to read all of the following page, especially* the very last section about opening issues or pull requests. https://github.com/moble/quaternion/wiki/Euler-angles-are-horrible

### Parameters

• q: array of quaternions, quaternion

The quaternion(s) need not be normalized, but must all be nonzero

### Returns

• alpha_beta_gamma: float array

Output shape is q.shape+(3,). These represent the angles (alpha, beta, gamma) in radians, where the normalized input quaternion represents exp(alpha*z/2) * exp(beta*y/2) * exp(gamma*z/2).

## as_float_array

as_float_array(a)


Source: quaternion/__init__.py

View the quaternion array as an array of floats This function is fast (of order 1 microsecond) because no data is copied; the returned quantity is just a "view" of the original. The output view has one more dimension (of size 4) than the input array, but is otherwise the same shape. The components along that last dimension represent the scalar and vector components of each quaternion in that order: w, x, y, z.

## as_quat_array

as_quat_array(a)


Source: quaternion/__init__.py

View a float array as an array of quaternions The input array must have a final dimension whose size is divisible by four (or better yet is 4), because successive indices in that last dimension will be considered successive components of the output quaternion. Each set of 4 components will be interpreted as the scalar and vector components of a quaternion in that order: w, x, y, z. This function is usually fast (of order 1 microsecond) because no data is copied; the returned quantity is just a "view" of the original. However, if the input array is not C-contiguous (basically, as you increment the index into the last dimension of the array, you just move to the neighboring float in memory), the data will need to be copied which may be quite slow. Therefore, you should try to ensure that the input array is in that order. Slices and transpositions will frequently break that rule. We will not convert back from a two-spinor array because there is no unique convention for them, so I don't want to mess with that. Also, we want to discourage users from the slow, memory-copying process of swapping columns required for useful definitions of the two-spinors.

## as_rotation_matrix

as_rotation_matrix(q)


Source: quaternion/__init__.py

Convert input quaternion to 3x3 rotation matrix For any quaternion q, this function returns a matrix m such that, for every vector v, we have m @ v.vec == q * v * q.conjugate() Here, @ is the standard python matrix multiplication operator and v.vec is the 3-vector part of the quaternion v.

### Parameters

• q: array of quaternions, quaternion

The quaternion(s) need not be normalized, but must all be nonzero

### Returns

• m: float array

Output shape is q.shape+(3,3). This matrix should multiply (from the left) a column vector to produce the rotated column vector.

## as_rotation_vector

as_rotation_vector(q)


Source: quaternion/__init__.py

Convert input quaternion to the axis-angle representation Note that if any of the input quaternions has norm zero, no error is raised, but NaNs will appear in the output.

### Parameters

• q: array of quaternions, quaternion

The quaternion(s) need not be normalized, but must all be nonzero

### Returns

• rot: float array

Output shape is q.shape+(3,). Each vector represents the axis of the rotation, with norm proportional to the angle of the rotation in radians.

## as_spherical_coords

as_spherical_coords(q)


Source: quaternion/__init__.py

Return the spherical coordinates corresponding to this quaternion Obviously, spherical coordinates do not contain as much information as a quaternion, so this function does lose some information. However, the returned spherical coordinates will represent the point(s) on the sphere to which the input quaternion(s) rotate the z axis.

### Parameters

• q: array of quaternions, quaternion

The quaternion(s) need not be normalized, but must be nonzero

### Returns

• vartheta_varphi: float array

Output shape is q.shape+(2,). These represent the angles (vartheta, varphi) in radians, where the normalized input quaternion represents exp(varphi*z/2) * exp(vartheta*y/2), up to an arbitrary inital rotation about z.

## as_spinor_array

as_spinor_array(a)


Source: quaternion/__init__.py

View a quaternion array as spinors in two-complex representation This function is relatively slow and scales poorly, because memory copying is apparently involved -- I think it's due to the "advanced indexing" required to swap the columns.

## as_vector_part

as_vector_part(q)


Source: quaternion/__init__.py

Create an array of vector parts from an array of quaternions.

### Parameters

• q: quaternion array_like

Array of quaternions.

### Returns

• v: array

Float array of shape q.shape + (3,)

## from_euler_angles

from_euler_angles(alpha_beta_gamma, beta=None, gamma=None)


Source: quaternion/__init__.py

Improve your life drastically Assumes the Euler angles correspond to the quaternion R via R = exp(alphaz/2) * exp(betay/2) * exp(gammaz/2) The angles naturally must be in radians for this to make any sense. NOTE: Before opening an issue reporting something "wrong" with this function, be sure to read all of the following page, especially* the very last section about opening issues or pull requests. https://github.com/moble/quaternion/wiki/Euler-angles-are-horrible

### Parameters

• alpha_beta_gamma: array of floats, float

This argument may either contain an array with last dimension of size 3, where those three elements describe the (alpha, beta, gamma) radian values for each rotation; or it may contain just the alpha values, in which case the next two arguments must also be given.

• beta: or array of floats, float, None

If this array is given, it must be able to broadcast against the first and third arguments.

• gamma: or array of floats, float, None

If this array is given, it must be able to broadcast against the first and second arguments.

### Returns

• R: quaternion array

The shape of this array will be the same as the input, except that the last dimension will be removed.

## from_float_array

from_float_array(a)


Source: quaternion/__init__.py

## from_rotation_matrix

from_rotation_matrix(rot, nonorthogonal=True)


Source: quaternion/__init__.py

Convert input 3x3 rotation matrix to unit quaternion For any orthogonal matrix rot, this function returns a quaternion q such that, for every pure-vector quaternion v, we have q * v * q.conjugate() == rot @ v.vec Here, @ is the standard python matrix multiplication operator and v.vec is the 3-vector part of the quaternion v. If rot is not orthogonal the "closest" orthogonal matrix is used; see Notes below.

### Parameters

• rot: 3) float array, 3, N, (...

Each 3x3 matrix represents a rotation by multiplying (from the left) a column vector to produce a rotated column vector. Note that this input may actually have ndims>3; it is just assumed that the last two dimensions have size 3, representing the matrix.

• nonorthogonal: optional, bool

If scipy.linalg is available, use the more robust algorithm of Bar-Itzhack. Default value is True.

### Returns

• q: array of quaternions

Unit quaternions resulting in rotations corresponding to input rotations. Output shape is rot.shape[:-2].

!!! note By default, if scipy.linalg is available, this function uses Bar-Itzhack's algorithm to allow for non-orthogonal matrices. [J. Guidance, Vol. 23, No. 6, p. 1085 http://dx.doi.org/10.2514/2.4654] This will almost certainly be quite a bit slower than simpler versions, though it will be more robust to numerical errors in the rotation matrix. Also note that Bar-Itzhack uses some pretty weird conventions. The last component of the quaternion appears to represent the scalar, and the quaternion itself is conjugated relative to the convention used throughout this module. If scipy.linalg is not available or if the optional nonorthogonal parameter is set to False, this function falls back to the possibly faster, but less robust, algorithm of Markley [J. Guidance, Vol. 31, No. 2, p. 440 http://dx.doi.org/10.2514/1.31730].

## from_rotation_vector

from_rotation_vector(rot)


Source: quaternion/__init__.py

Convert input 3-vector in axis-angle representation to unit quaternion

### Parameters

• rot: (Nx3) float array

Each vector represents the axis of the rotation, with norm proportional to the angle of the rotation in radians.

### Returns

• q: array of quaternions

Unit quaternions resulting in rotations corresponding to input rotations. Output shape is rot.shape[:-1].

## from_spherical_coords

from_spherical_coords(theta_phi, phi=None)


Source: quaternion/__init__.py

Return the quaternion corresponding to these spherical coordinates Assumes the spherical coordinates correspond to the quaternion R via R = exp(phiz/2) * exp(thetay/2) The angles naturally must be in radians for this to make any sense. Note that this quaternion rotates z onto the point with the given spherical coordinates, but also rotates x and y onto the usual basis vectors (theta and phi, respectively) at that point.

### Parameters

• theta_phi: array of floats, float

This argument may either contain an array with last dimension of size 2, where those two elements describe the (theta, phi) values in radians for each point; or it may contain just the theta values in radians, in which case the next argument must also be given.

• phi: or array of floats, float, None

If this array is given, it must be able to broadcast against the first argument.

### Returns

• R: quaternion array

If the second argument is not given to this function, the shape will be the same as the input shape except for the last dimension, which will be removed. If the second argument is given, this output array will have the shape resulting from broadcasting the two input arrays against each other.

## from_vector_part

from_vector_part(v, vector_axis=-1)


Source: quaternion/__init__.py

Create a quaternion array from an array of the vector parts. Essentially, this just inserts a 0 in front of each vector part, and re-interprets the result as a quaternion.

### Parameters

• v: array_like

Array of vector parts of quaternions. When interpreted as a numpy array, if the dtype is quaternion, the array is returned immediately, and the following argument is ignored. Otherwise, it it must be a float array with dimension vector_axis of size 3 or 4.

• vector_axis: optional, int

The axis to interpret as containing the vector components. The default is -1.

### Returns

• q: array of quaternions

Quaternions with vector parts corresponding to input vectors.

## isclose

isclose(a, b, rtol=8.881784197001252e-16, atol=0.0, equal_nan=False)


Source: quaternion/__init__.py

Returns a boolean array where two arrays are element-wise equal within a tolerance. This function is essentially a copy of the numpy.isclose function, with different default tolerances and one minor changes necessary to deal correctly with quaternions. The tolerance values are positive, typically very small numbers. The relative difference (rtol * abs(b)) and the absolute difference atol are added together to compare against the absolute difference between a and b.

### Parameters

• a, b: array_like

Input arrays to compare.

• rtol: float

The relative tolerance parameter (see Notes).

• atol: float

The absolute tolerance parameter (see Notes).

• equal_nan: bool

Whether to compare NaN's as equal. If True, NaN's in a will be considered equal to NaN's in b in the output array.

### Returns

• y: array_like

Returns a boolean array of where a and b are equal within the given tolerance. If both a and b are scalars, returns a single boolean value.

### Examples

quaternion.isclose([1e10*quaternion.x, 1e-7*quaternion.y], [1.00001e10*quaternion.x, 1e-8*quaternion.y],
...     rtol=1.e-5, atol=1.e-8)
array([True, False])
quaternion.isclose([1e10*quaternion.x, 1e-8*quaternion.y], [1.00001e10*quaternion.x, 1e-9*quaternion.y],
...     rtol=1.e-5, atol=1.e-8)
array([True, True])
quaternion.isclose([1e10*quaternion.x, 1e-8*quaternion.y], [1.0001e10*quaternion.x, 1e-9*quaternion.y],
...     rtol=1.e-5, atol=1.e-8)
array([False, True])
quaternion.isclose([quaternion.x, np.nan*quaternion.y], [quaternion.x, np.nan*quaternion.y])
array([True, False])
quaternion.isclose([quaternion.x, np.nan*quaternion.y], [quaternion.x, np.nan*quaternion.y], equal_nan=True)
array([True, True])



!!! note For finite values, isclose uses the following equation to test whether two floating point values are equivalent: absolute(a - b) <= (atol + rtol * absolute(b)) The above equation is not symmetric in a and b, so that isclose(a, b) might be different from isclose(b, a) in some rare cases.

## rotate_vectors

rotate_vectors(R, v, axis=-1)


Source: quaternion/__init__.py

Rotate vectors by given quaternions This function is for the case where each quaternion (possibly the only input quaternion) is used to rotate multiple vectors. If each quaternion is only rotating a single vector, it is more efficient to use the standard formula vprime = R * v * R.conjugate() (Note that from_vector_part and as_vector_part may be helpful.)

### Parameters

• R: quaternion array

Quaternions by which to rotate the input vectors

• v: float array

Three-vectors to be rotated.

• axis: int

Axis of the v array to use as the vector dimension. This axis of v must have length 3.

### Returns

• vprime: float array

The rotated vectors. This array has shape R.shape+v.shape.

!!! note For simplicity, this function converts the input quaternion(s) to matrix form, and rotates the input vector(s) by the usual matrix multiplication. As noted above, if each input quaternion is only used to rotate a single vector, this is not the most efficient approach. The simple formula shown above is faster than this function, though it should be noted that the most efficient approach (in terms of operation counts) is to use the formula v' = v + 2 * r x (s * v + r x v) / m where x represents the cross product, s and r are the scalar and vector parts of the quaternion, respectively, and m is the sum of the squares of the components of the quaternion. If you are looping over a very large number of quaternions, and just rotating a single vector each time, you might want to implement that alternative algorithm using numba (or something that doesn't use python).