transformations¶
Homogeneous Transformation Matrices and Quaternions.
A library for calculating 4x4 matrices for translating, rotating, reflecting, scaling, shearing, projecting, orthogonalizing, and superimposing arrays of 3D homogeneous coordinates as well as for converting between rotation matrices, Euler angles, and quaternions. Also includes an Arcball control object and functions to decompose transformation matrices.
Authors: | Christoph Gohlke, Laboratory for Fluorescence Dynamics, University of California, Irvine |
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Version: | 20090418 |
Requirements¶
- Python 2.6
- Numpy 1.3
- transformations.c 20090418 (optional implementation of some functions in C)
Notes¶
Matrices (M) can be inverted using numpy.linalg.inv(M), concatenated using numpy.dot(M0, M1), or used to transform homogeneous coordinates (v) using numpy.dot(M, v) for shape (4, *) “point of arrays”, respectively numpy.dot(v, M.T) for shape (*, 4) “array of points”.
Calculations are carried out with numpy.float64 precision.
This Python implementation is not optimized for speed.
Vector, point, quaternion, and matrix function arguments are expected to be “array like”, i.e. tuple, list, or numpy arrays.
Return types are numpy arrays unless specified otherwise.
Angles are in radians unless specified otherwise.
Quaternions ix+jy+kz+w are represented as [x, y, z, w].
Use the transpose of transformation matrices for OpenGL glMultMatrixd().
A triple of Euler angles can be applied/interpreted in 24 ways, which can be specified using a 4 character string or encoded 4-tuple:
Axes 4-string: e.g. ‘sxyz’ or ‘ryxy’
- first character : rotations are applied to ‘s’tatic or ‘r’otating frame
- remaining characters : successive rotation axis ‘x’, ‘y’, or ‘z’
Axes 4-tuple: e.g. (0, 0, 0, 0) or (1, 1, 1, 1)
- inner axis: code of axis (‘x’:0, ‘y’:1, ‘z’:2) of rightmost matrix.
- parity : even (0) if inner axis ‘x’ is followed by ‘y’, ‘y’ is followed by ‘z’, or ‘z’ is followed by ‘x’. Otherwise odd (1).
- repetition : first and last axis are same (1) or different (0).
- frame : rotations are applied to static (0) or rotating (1) frame.
References¶
- Matrices and transformations. Ronald Goldman. In “Graphics Gems I”, pp 472-475. Morgan Kaufmann, 1990.
- More matrices and transformations: shear and pseudo-perspective. Ronald Goldman. In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.
- Decomposing a matrix into simple transformations. Spencer Thomas. In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.
- Recovering the data from the transformation matrix. Ronald Goldman. In “Graphics Gems II”, pp 324-331. Morgan Kaufmann, 1991.
- Euler angle conversion. Ken Shoemake. In “Graphics Gems IV”, pp 222-229. Morgan Kaufmann, 1994.
- Arcball rotation control. Ken Shoemake. In “Graphics Gems IV”, pp 175-192. Morgan Kaufmann, 1994.
- Representing attitude: Euler angles, unit quaternions, and rotation vectors. James Diebel. 2006.
- A discussion of the solution for the best rotation to relate two sets of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828.
- Closed-form solution of absolute orientation using unit quaternions. BKP Horn. J Opt Soc Am A. 1987. 4(4), 629-642.
- Quaternions. Ken Shoemake. http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf
- From quaternion to matrix and back. JMP van Waveren. 2005. http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm
- Uniform random rotations. Ken Shoemake. In “Graphics Gems III”, pp 124-132. Morgan Kaufmann, 1992.
Examples¶
>>> alpha, beta, gamma = 0.123, -1.234, 2.345
>>> origin, xaxis, yaxis, zaxis = (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)
>>> I = identity_matrix()
>>> Rx = rotation_matrix(alpha, xaxis)
>>> Ry = rotation_matrix(beta, yaxis)
>>> Rz = rotation_matrix(gamma, zaxis)
>>> R = concatenate_matrices(Rx, Ry, Rz)
>>> euler = euler_from_matrix(R, 'rxyz')
>>> numpy.allclose([alpha, beta, gamma], euler)
True
>>> Re = euler_matrix(alpha, beta, gamma, 'rxyz')
>>> is_same_transform(R, Re)
True
>>> al, be, ga = euler_from_matrix(Re, 'rxyz')
>>> is_same_transform(Re, euler_matrix(al, be, ga, 'rxyz'))
True
>>> qx = quaternion_about_axis(alpha, xaxis)
>>> qy = quaternion_about_axis(beta, yaxis)
>>> qz = quaternion_about_axis(gamma, zaxis)
>>> q = quaternion_multiply(qx, qy)
>>> q = quaternion_multiply(q, qz)
>>> Rq = quaternion_matrix(q)
>>> is_same_transform(R, Rq)
True
>>> S = scale_matrix(1.23, origin)
>>> T = translation_matrix((1, 2, 3))
>>> Z = shear_matrix(beta, xaxis, origin, zaxis)
>>> R = random_rotation_matrix(numpy.random.rand(3))
>>> M = concatenate_matrices(T, R, Z, S)
>>> scale, shear, angles, trans, persp = decompose_matrix(M)
>>> numpy.allclose(scale, 1.23)
True
>>> numpy.allclose(trans, (1, 2, 3))
True
>>> numpy.allclose(shear, (0, math.tan(beta), 0))
True
>>> is_same_transform(R, euler_matrix(axes='sxyz', *angles))
True
>>> M1 = compose_matrix(scale, shear, angles, trans, persp)
>>> is_same_transform(M, M1)
True
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class
tf.transformations.
Arcball
(initial=None)¶ Virtual Trackball Control.
>>> ball = Arcball() >>> ball = Arcball(initial=numpy.identity(4)) >>> ball.place([320, 320], 320) >>> ball.down([500, 250]) >>> ball.drag([475, 275]) >>> R = ball.matrix() >>> numpy.allclose(numpy.sum(R), 3.90583455) True >>> ball = Arcball(initial=[0, 0, 0, 1]) >>> ball.place([320, 320], 320) >>> ball.setaxes([1,1,0], [-1, 1, 0]) >>> ball.setconstrain(True) >>> ball.down([400, 200]) >>> ball.drag([200, 400]) >>> R = ball.matrix() >>> numpy.allclose(numpy.sum(R), 0.2055924) True >>> ball.next()
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down
(point)¶ Set initial cursor window coordinates and pick constrain-axis.
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drag
(point)¶ Update current cursor window coordinates.
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getconstrain
()¶ Return state of constrain to axis mode.
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matrix
()¶ Return homogeneous rotation matrix.
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next
(acceleration=0.0)¶ Continue rotation in direction of last drag.
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place
(center, radius)¶ Place Arcball, e.g. when window size changes.
- center : sequence[2]
- Window coordinates of trackball center.
- radius : float
- Radius of trackball in window coordinates.
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setaxes
(*axes)¶ Set axes to constrain rotations.
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setconstrain
(constrain)¶ Set state of constrain to axis mode.
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tf.transformations.
arcball_constrain_to_axis
(point, axis)¶ Return sphere point perpendicular to axis.
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tf.transformations.
arcball_map_to_sphere
(point, center, radius)¶ Return unit sphere coordinates from window coordinates.
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tf.transformations.
arcball_nearest_axis
(point, axes)¶ Return axis, which arc is nearest to point.
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tf.transformations.
clip_matrix
(left, right, bottom, top, near, far, perspective=False)¶ Return matrix to obtain normalized device coordinates from frustrum.
The frustrum bounds are axis-aligned along x (left, right), y (bottom, top) and z (near, far).
Normalized device coordinates are in range [-1, 1] if coordinates are inside the frustrum.
If perspective is True the frustrum is a truncated pyramid with the perspective point at origin and direction along z axis, otherwise an orthographic canonical view volume (a box).
Homogeneous coordinates transformed by the perspective clip matrix need to be dehomogenized (devided by w coordinate).
>>> frustrum = numpy.random.rand(6) >>> frustrum[1] += frustrum[0] >>> frustrum[3] += frustrum[2] >>> frustrum[5] += frustrum[4] >>> M = clip_matrix(*frustrum, perspective=False) >>> numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0]) array([-1., -1., -1., 1.]) >>> numpy.dot(M, [frustrum[1], frustrum[3], frustrum[5], 1.0]) array([ 1., 1., 1., 1.]) >>> M = clip_matrix(*frustrum, perspective=True) >>> v = numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0]) >>> v / v[3] array([-1., -1., -1., 1.]) >>> v = numpy.dot(M, [frustrum[1], frustrum[3], frustrum[4], 1.0]) >>> v / v[3] array([ 1., 1., -1., 1.])
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tf.transformations.
compose_matrix
(scale=None, shear=None, angles=None, translate=None, perspective=None)¶ Return transformation matrix from sequence of transformations.
This is the inverse of the decompose_matrix function.
- Sequence of transformations:
- scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix
>>> scale = numpy.random.random(3) - 0.5 >>> shear = numpy.random.random(3) - 0.5 >>> angles = (numpy.random.random(3) - 0.5) * (2*math.pi) >>> trans = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(4) - 0.5 >>> M0 = compose_matrix(scale, shear, angles, trans, persp) >>> result = decompose_matrix(M0) >>> M1 = compose_matrix(*result) >>> is_same_transform(M0, M1) True
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tf.transformations.
concatenate_matrices
(*matrices)¶ Return concatenation of series of transformation matrices.
>>> M = numpy.random.rand(16).reshape((4, 4)) - 0.5 >>> numpy.allclose(M, concatenate_matrices(M)) True >>> numpy.allclose(numpy.dot(M, M.T), concatenate_matrices(M, M.T)) True
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tf.transformations.
decompose_matrix
(matrix)¶ Return sequence of transformations from transformation matrix.
- matrix : array_like
- Non-degenerative homogeneous transformation matrix
- Return tuple of:
- scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix
Raise ValueError if matrix is of wrong type or degenerative.
>>> T0 = translation_matrix((1, 2, 3)) >>> scale, shear, angles, trans, persp = decompose_matrix(T0) >>> T1 = translation_matrix(trans) >>> numpy.allclose(T0, T1) True >>> S = scale_matrix(0.123) >>> scale, shear, angles, trans, persp = decompose_matrix(S) >>> scale[0] 0.123 >>> R0 = euler_matrix(1, 2, 3) >>> scale, shear, angles, trans, persp = decompose_matrix(R0) >>> R1 = euler_matrix(*angles) >>> numpy.allclose(R0, R1) True
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tf.transformations.
euler_from_matrix
(matrix, axes='sxyz')¶ Return Euler angles from rotation matrix for specified axis sequence.
axes : One of 24 axis sequences as string or encoded tuple
Note that many Euler angle triplets can describe one matrix.
>>> R0 = euler_matrix(1, 2, 3, 'syxz') >>> al, be, ga = euler_from_matrix(R0, 'syxz') >>> R1 = euler_matrix(al, be, ga, 'syxz') >>> numpy.allclose(R0, R1) True >>> angles = (4.0*math.pi) * (numpy.random.random(3) - 0.5) >>> for axes in _AXES2TUPLE.keys(): ... R0 = euler_matrix(axes=axes, *angles) ... R1 = euler_matrix(axes=axes, *euler_from_matrix(R0, axes)) ... if not numpy.allclose(R0, R1): print axes, "failed"
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tf.transformations.
euler_from_quaternion
(quaternion, axes='sxyz')¶ Return Euler angles from quaternion for specified axis sequence.
>>> angles = euler_from_quaternion([0.06146124, 0, 0, 0.99810947]) >>> numpy.allclose(angles, [0.123, 0, 0]) True
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tf.transformations.
euler_matrix
(ai, aj, ak, axes='sxyz')¶ Return homogeneous rotation matrix from Euler angles and axis sequence.
ai, aj, ak : Euler’s roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple
>>> R = euler_matrix(1, 2, 3, 'syxz') >>> numpy.allclose(numpy.sum(R[0]), -1.34786452) True >>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1)) >>> numpy.allclose(numpy.sum(R[0]), -0.383436184) True >>> ai, aj, ak = (4.0*math.pi) * (numpy.random.random(3) - 0.5) >>> for axes in _AXES2TUPLE.keys(): ... R = euler_matrix(ai, aj, ak, axes) >>> for axes in _TUPLE2AXES.keys(): ... R = euler_matrix(ai, aj, ak, axes)
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tf.transformations.
identity_matrix
()¶ Return 4x4 identity/unit matrix.
>>> I = identity_matrix() >>> numpy.allclose(I, numpy.dot(I, I)) True >>> numpy.sum(I), numpy.trace(I) (4.0, 4.0) >>> numpy.allclose(I, numpy.identity(4, dtype=numpy.float64)) True
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tf.transformations.
inverse_matrix
(matrix)¶ Return inverse of square transformation matrix.
>>> M0 = random_rotation_matrix() >>> M1 = inverse_matrix(M0.T) >>> numpy.allclose(M1, numpy.linalg.inv(M0.T)) True >>> for size in range(1, 7): ... M0 = numpy.random.rand(size, size) ... M1 = inverse_matrix(M0) ... if not numpy.allclose(M1, numpy.linalg.inv(M0)): print size
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tf.transformations.
is_same_transform
(matrix0, matrix1)¶ Return True if two matrices perform same transformation.
>>> is_same_transform(numpy.identity(4), numpy.identity(4)) True >>> is_same_transform(numpy.identity(4), random_rotation_matrix()) False
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tf.transformations.
orthogonalization_matrix
(lengths, angles)¶ Return orthogonalization matrix for crystallographic cell coordinates.
Angles are expected in degrees.
The de-orthogonalization matrix is the inverse.
>>> O = orthogonalization_matrix((10., 10., 10.), (90., 90., 90.)) >>> numpy.allclose(O[:3, :3], numpy.identity(3, float) * 10) True >>> O = orthogonalization_matrix([9.8, 12.0, 15.5], [87.2, 80.7, 69.7]) >>> numpy.allclose(numpy.sum(O), 43.063229) True
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tf.transformations.
projection_from_matrix
(matrix, pseudo=False)¶ Return projection plane and perspective point from projection matrix.
Return values are same as arguments for projection_matrix function: point, normal, direction, perspective, and pseudo.
>>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(3) - 0.5 >>> P0 = projection_matrix(point, normal) >>> result = projection_from_matrix(P0) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, direct) >>> result = projection_from_matrix(P0) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=False) >>> result = projection_from_matrix(P0, pseudo=False) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=True) >>> result = projection_from_matrix(P0, pseudo=True) >>> P1 = projection_matrix(*result) >>> is_same_transform(P0, P1) True
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tf.transformations.
projection_matrix
(point, normal, direction=None, perspective=None, pseudo=False)¶ Return matrix to project onto plane defined by point and normal.
Using either perspective point, projection direction, or none of both.
If pseudo is True, perspective projections will preserve relative depth such that Perspective = dot(Orthogonal, PseudoPerspective).
>>> P = projection_matrix((0, 0, 0), (1, 0, 0)) >>> numpy.allclose(P[1:, 1:], numpy.identity(4)[1:, 1:]) True >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> persp = numpy.random.random(3) - 0.5 >>> P0 = projection_matrix(point, normal) >>> P1 = projection_matrix(point, normal, direction=direct) >>> P2 = projection_matrix(point, normal, perspective=persp) >>> P3 = projection_matrix(point, normal, perspective=persp, pseudo=True) >>> is_same_transform(P2, numpy.dot(P0, P3)) True >>> P = projection_matrix((3, 0, 0), (1, 1, 0), (1, 0, 0)) >>> v0 = (numpy.random.rand(4, 5) - 0.5) * 20.0 >>> v0[3] = 1.0 >>> v1 = numpy.dot(P, v0) >>> numpy.allclose(v1[1], v0[1]) True >>> numpy.allclose(v1[0], 3.0-v1[1]) True
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tf.transformations.
quaternion_about_axis
(angle, axis)¶ Return quaternion for rotation about axis.
>>> q = quaternion_about_axis(0.123, (1, 0, 0)) >>> numpy.allclose(q, [0.06146124, 0, 0, 0.99810947]) True
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tf.transformations.
quaternion_conjugate
(quaternion)¶ Return conjugate of quaternion.
>>> q0 = random_quaternion() >>> q1 = quaternion_conjugate(q0) >>> q1[3] == q0[3] and all(q1[:3] == -q0[:3]) True
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tf.transformations.
quaternion_from_euler
(ai, aj, ak, axes='sxyz')¶ Return quaternion from Euler angles and axis sequence.
ai, aj, ak : Euler’s roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple
>>> q = quaternion_from_euler(1, 2, 3, 'ryxz') >>> numpy.allclose(q, [0.310622, -0.718287, 0.444435, 0.435953]) True
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tf.transformations.
quaternion_from_matrix
(matrix)¶ Return quaternion from rotation matrix.
>>> R = rotation_matrix(0.123, (1, 2, 3)) >>> q = quaternion_from_matrix(R) >>> numpy.allclose(q, [0.0164262, 0.0328524, 0.0492786, 0.9981095]) True
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tf.transformations.
quaternion_inverse
(quaternion)¶ Return inverse of quaternion.
>>> q0 = random_quaternion() >>> q1 = quaternion_inverse(q0) >>> numpy.allclose(quaternion_multiply(q0, q1), [0, 0, 0, 1]) True
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tf.transformations.
quaternion_matrix
(quaternion)¶ Return homogeneous rotation matrix from quaternion.
>>> R = quaternion_matrix([0.06146124, 0, 0, 0.99810947]) >>> numpy.allclose(R, rotation_matrix(0.123, (1, 0, 0))) True
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tf.transformations.
quaternion_multiply
(quaternion1, quaternion0)¶ Return multiplication of two quaternions.
>>> q = quaternion_multiply([1, -2, 3, 4], [-5, 6, 7, 8]) >>> numpy.allclose(q, [-44, -14, 48, 28]) True
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tf.transformations.
quaternion_slerp
(quat0, quat1, fraction, spin=0, shortestpath=True)¶ Return spherical linear interpolation between two quaternions.
>>> q0 = random_quaternion() >>> q1 = random_quaternion() >>> q = quaternion_slerp(q0, q1, 0.0) >>> numpy.allclose(q, q0) True >>> q = quaternion_slerp(q0, q1, 1.0, 1) >>> numpy.allclose(q, q1) True >>> q = quaternion_slerp(q0, q1, 0.5) >>> angle = math.acos(numpy.dot(q0, q)) >>> numpy.allclose(2.0, math.acos(numpy.dot(q0, q1)) / angle) or numpy.allclose(2.0, math.acos(-numpy.dot(q0, q1)) / angle) True
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tf.transformations.
random_quaternion
(rand=None)¶ Return uniform random unit quaternion.
- rand: array like or None
- Three independent random variables that are uniformly distributed between 0 and 1.
>>> q = random_quaternion() >>> numpy.allclose(1.0, vector_norm(q)) True >>> q = random_quaternion(numpy.random.random(3)) >>> q.shape (4,)
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tf.transformations.
random_rotation_matrix
(rand=None)¶ Return uniform random rotation matrix.
- rnd: array like
- Three independent random variables that are uniformly distributed between 0 and 1 for each returned quaternion.
>>> R = random_rotation_matrix() >>> numpy.allclose(numpy.dot(R.T, R), numpy.identity(4)) True
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tf.transformations.
random_vector
(size)¶ Return array of random doubles in the half-open interval [0.0, 1.0).
>>> v = random_vector(10000) >>> numpy.all(v >= 0.0) and numpy.all(v < 1.0) True >>> v0 = random_vector(10) >>> v1 = random_vector(10) >>> numpy.any(v0 == v1) False
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tf.transformations.
reflection_from_matrix
(matrix)¶ Return mirror plane point and normal vector from reflection matrix.
>>> v0 = numpy.random.random(3) - 0.5 >>> v1 = numpy.random.random(3) - 0.5 >>> M0 = reflection_matrix(v0, v1) >>> point, normal = reflection_from_matrix(M0) >>> M1 = reflection_matrix(point, normal) >>> is_same_transform(M0, M1) True
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tf.transformations.
reflection_matrix
(point, normal)¶ Return matrix to mirror at plane defined by point and normal vector.
>>> v0 = numpy.random.random(4) - 0.5 >>> v0[3] = 1.0 >>> v1 = numpy.random.random(3) - 0.5 >>> R = reflection_matrix(v0, v1) >>> numpy.allclose(2., numpy.trace(R)) True >>> numpy.allclose(v0, numpy.dot(R, v0)) True >>> v2 = v0.copy() >>> v2[:3] += v1 >>> v3 = v0.copy() >>> v2[:3] -= v1 >>> numpy.allclose(v2, numpy.dot(R, v3)) True
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tf.transformations.
rotation_from_matrix
(matrix)¶ Return rotation angle and axis from rotation matrix.
>>> angle = (random.random() - 0.5) * (2*math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> angle, direc, point = rotation_from_matrix(R0) >>> R1 = rotation_matrix(angle, direc, point) >>> is_same_transform(R0, R1) True
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tf.transformations.
rotation_matrix
(angle, direction, point=None)¶ Return matrix to rotate about axis defined by point and direction.
>>> angle = (random.random() - 0.5) * (2*math.pi) >>> direc = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(angle-2*math.pi, direc, point) >>> is_same_transform(R0, R1) True >>> R0 = rotation_matrix(angle, direc, point) >>> R1 = rotation_matrix(-angle, -direc, point) >>> is_same_transform(R0, R1) True >>> I = numpy.identity(4, numpy.float64) >>> numpy.allclose(I, rotation_matrix(math.pi*2, direc)) True >>> numpy.allclose(2., numpy.trace(rotation_matrix(math.pi/2, ... direc, point))) True
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tf.transformations.
scale_from_matrix
(matrix)¶ Return scaling factor, origin and direction from scaling matrix.
>>> factor = random.random() * 10 - 5 >>> origin = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> S0 = scale_matrix(factor, origin) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True >>> S0 = scale_matrix(factor, origin, direct) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True
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tf.transformations.
scale_matrix
(factor, origin=None, direction=None)¶ Return matrix to scale by factor around origin in direction.
Use factor -1 for point symmetry.
>>> v = (numpy.random.rand(4, 5) - 0.5) * 20.0 >>> v[3] = 1.0 >>> S = scale_matrix(-1.234) >>> numpy.allclose(numpy.dot(S, v)[:3], -1.234*v[:3]) True >>> factor = random.random() * 10 - 5 >>> origin = numpy.random.random(3) - 0.5 >>> direct = numpy.random.random(3) - 0.5 >>> S = scale_matrix(factor, origin) >>> S = scale_matrix(factor, origin, direct)
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tf.transformations.
shear_from_matrix
(matrix)¶ Return shear angle, direction and plane from shear matrix.
>>> angle = (random.random() - 0.5) * 4*math.pi >>> direct = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.cross(direct, numpy.random.random(3)) >>> S0 = shear_matrix(angle, direct, point, normal) >>> angle, direct, point, normal = shear_from_matrix(S0) >>> S1 = shear_matrix(angle, direct, point, normal) >>> is_same_transform(S0, S1) True
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tf.transformations.
shear_matrix
(angle, direction, point, normal)¶ Return matrix to shear by angle along direction vector on shear plane.
The shear plane is defined by a point and normal vector. The direction vector must be orthogonal to the plane’s normal vector.
A point P is transformed by the shear matrix into P” such that the vector P-P” is parallel to the direction vector and its extent is given by the angle of P-P’-P”, where P’ is the orthogonal projection of P onto the shear plane.
>>> angle = (random.random() - 0.5) * 4*math.pi >>> direct = numpy.random.random(3) - 0.5 >>> point = numpy.random.random(3) - 0.5 >>> normal = numpy.cross(direct, numpy.random.random(3)) >>> S = shear_matrix(angle, direct, point, normal) >>> numpy.allclose(1.0, numpy.linalg.det(S)) True
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tf.transformations.
superimposition_matrix
(v0, v1, scaling=False, usesvd=True)¶ Return matrix to transform given vector set into second vector set.
v0 and v1 are shape (3, *) or (4, *) arrays of at least 3 vectors.
If usesvd is True, the weighted sum of squared deviations (RMSD) is minimized according to the algorithm by W. Kabsch [8]. Otherwise the quaternion based algorithm by B. Horn [9] is used (slower when using this Python implementation).
The returned matrix performs rotation, translation and uniform scaling (if specified).
>>> v0 = numpy.random.rand(3, 10) >>> M = superimposition_matrix(v0, v0) >>> numpy.allclose(M, numpy.identity(4)) True >>> R = random_rotation_matrix(numpy.random.random(3)) >>> v0 = ((1,0,0), (0,1,0), (0,0,1), (1,1,1)) >>> v1 = numpy.dot(R, v0) >>> M = superimposition_matrix(v0, v1) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20.0 >>> v0[3] = 1.0 >>> v1 = numpy.dot(R, v0) >>> M = superimposition_matrix(v0, v1) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> S = scale_matrix(random.random()) >>> T = translation_matrix(numpy.random.random(3)-0.5) >>> M = concatenate_matrices(T, R, S) >>> v1 = numpy.dot(M, v0) >>> v0[:3] += numpy.random.normal(0.0, 1e-9, 300).reshape(3, -1) >>> M = superimposition_matrix(v0, v1, scaling=True) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False) >>> numpy.allclose(v1, numpy.dot(M, v0)) True >>> v = numpy.empty((4, 100, 3), dtype=numpy.float64) >>> v[:, :, 0] = v0 >>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False) >>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0])) True
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tf.transformations.
translation_from_matrix
(matrix)¶ Return translation vector from translation matrix.
>>> v0 = numpy.random.random(3) - 0.5 >>> v1 = translation_from_matrix(translation_matrix(v0)) >>> numpy.allclose(v0, v1) True
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tf.transformations.
translation_matrix
(direction)¶ Return matrix to translate by direction vector.
>>> v = numpy.random.random(3) - 0.5 >>> numpy.allclose(v, translation_matrix(v)[:3, 3]) True
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tf.transformations.
unit_vector
(data, axis=None, out=None)¶ Return ndarray normalized by length, i.e. eucledian norm, along axis.
>>> v0 = numpy.random.random(3) >>> v1 = unit_vector(v0) >>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0)) True >>> v0 = numpy.random.rand(5, 4, 3) >>> v1 = unit_vector(v0, axis=-1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2) >>> numpy.allclose(v1, v2) True >>> v1 = unit_vector(v0, axis=1) >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1) >>> numpy.allclose(v1, v2) True >>> v1 = numpy.empty((5, 4, 3), dtype=numpy.float64) >>> unit_vector(v0, axis=1, out=v1) >>> numpy.allclose(v1, v2) True >>> list(unit_vector([])) [] >>> list(unit_vector([1.0])) [1.0]
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tf.transformations.
vector_norm
(data, axis=None, out=None)¶ Return length, i.e. eucledian norm, of ndarray along axis.
>>> v = numpy.random.random(3) >>> n = vector_norm(v) >>> numpy.allclose(n, numpy.linalg.norm(v)) True >>> v = numpy.random.rand(6, 5, 3) >>> n = vector_norm(v, axis=-1) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=2))) True >>> n = vector_norm(v, axis=1) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) True >>> v = numpy.random.rand(5, 4, 3) >>> n = numpy.empty((5, 3), dtype=numpy.float64) >>> vector_norm(v, axis=1, out=n) >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) True >>> vector_norm([]) 0.0 >>> vector_norm([1.0]) 1.0