axangles

Functions for working with axis, angle rotations

See quaternions for conversions between axis, angle pairs and quaternions.

Terms used in function names:

• mat : array shape (3, 3) (3D non-homogenous coordinates)

• aff : affine array shape (4, 4) (3D homogenous coordinates)

• axangle : rotations encoded by axis vector and angle scalar

aff2axangle(aff)	Return axis, angle and point from affine
axangle2aff(axis, angle[, point])	Return affine encoding rotation by angle about axis.
axangle2mat(axis, angle[, is_normalized])	Rotation matrix for rotation angle angle around axis
mat2axangle(mat[, unit_thresh])	Return axis, angle and point from (3, 3) matrix mat

aff2axangle

transforms3d.axangles.aff2axangle(aff)

Return axis, angle and point from affine

Parameters

affarray-like shape (4,4)

Returns

axisarray shape (3,)

vector giving axis of rotation

anglescalar

angle of rotation in radians.

pointarray shape (3,)

point around which rotation is performed

Notes

http://en.wikipedia.org/wiki/Rotation_matrix#Axis_of_a_rotation

Examples

>>> direc = np.random.random(3) - 0.5
>>> angle = (np.random.random() - 0.5) * (2*math.pi)
>>> point = np.random.random(3) - 0.5
>>> R0 = axangle2aff(direc, angle, point)
>>> direc, angle, point = aff2axangle(R0)
>>> R1 = axangle2aff(direc, angle, point)
>>> np.allclose(R0, R1)
True

axangle2aff

transforms3d.axangles.axangle2aff(axis, angle, point=None)

Return affine encoding rotation by angle about axis.

Parameters

axisarray shape (3,)

vector giving axis of rotation

anglescalar

angle of rotation, in radians.

Returns

Aarray shape (4, 4)

Affine array to be multiplied on left of coordinate column vector to apply given rotation.

Notes

Applying a rotation around a point is the same as applying a translation of -point to move point to the origin, rotating, then applying a translation of point. If R is the rotation matrix, than the affine for the rotation about point P is:

[R00, R01, R02, P0 - P0*R00 - P1*R01 - P2*R02]
[R10, R11, R12, P1 - P0*R10 - P1*R11 - P2*R12]
[R20, R21, R22, P2 - P0*R20 - P1*R21 - P2*R22]
[  0,   0,   0,                             1]

(see derivations)

Examples

>>> angle = (np.random.random() - 0.5) * (2*math.pi)
>>> direc = np.random.random(3) - 0.5
>>> point = np.random.random(3) - 0.5
>>> R0 = axangle2aff(direc, angle, point)
>>> R1 = axangle2aff(direc, angle-2*math.pi, point)
>>> np.allclose(R0, R1)
True
>>> R0 = axangle2aff(direc, angle, point)
>>> R1 = axangle2aff(-direc, -angle, point)
>>> np.allclose(R0, R1)
True
>>> I = np.identity(4, np.float64)
>>> np.allclose(I, axangle2aff(direc, math.pi*2))
True
>>> np.allclose(2., np.trace(axangle2aff(direc,
...                                      math.pi/2,
...                                      point)))
True

axangle2mat

transforms3d.axangles.axangle2mat(axis, angle, is_normalized=False)

Rotation matrix for rotation angle angle around axis

Parameters

axis3 element sequence

vector specifying axis for rotation.

anglescalar

angle of rotation in radians.

is_normalizedbool, optional

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

Returns

matarray shape (3,3)

rotation matrix for specified rotation

Notes

From: http://en.wikipedia.org/wiki/Rotation_matrix#Axis_and_angle

mat2axangle

transforms3d.axangles.mat2axangle(mat, unit_thresh=1e-05)

Return axis, angle and point from (3, 3) matrix mat

Parameters

matarray-like shape (3, 3)

Rotation matrix

unit_threshfloat, optional

Tolerable difference from 1 when testing for unit eigenvalues to confirm mat is a rotation matrix.

Returns

axisarray shape (3,)

vector giving axis of rotation

anglescalar

angle of rotation in radians.

Notes

http://en.wikipedia.org/wiki/Rotation_matrix#Axis_of_a_rotation

Examples

>>> direc = np.random.random(3) - 0.5
>>> angle = (np.random.random() - 0.5) * (2*math.pi)
>>> R0 = axangle2mat(direc, angle)
>>> direc, angle = mat2axangle(R0)
>>> R1 = axangle2mat(direc, angle)
>>> np.allclose(R0, R1)
True