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: | 2012.10.14 |
The API is not stable yet and is expected to change between revisions.
This Python code is not optimized for speed. Refer to the transformations.c module for a faster implementation of some functions.
Documentation in HTML format can be generated with epydoc.
Matrices (M) can be inverted using numpy.linalg.inv(M), be concatenated using numpy.dot(M0, M1), or transform homogeneous coordinate arrays (v) using numpy.dot(M, v) for shape (4, *) column vectors, respectively numpy.dot(v, M.T) for shape (*, 4) row vectors (“array of points”).
This module follows the “column vectors on the right” and “row major storage” (C contiguous) conventions. The translation components are in the right column of the transformation matrix, i.e. M[:3, 3]. The transpose of the transformation matrices may have to be used to interface with other graphics systems, e.g. with OpenGL’s glMultMatrixd(). See also [16].
Calculations are carried out with numpy.float64 precision.
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 w+ix+jy+kz are represented as [w, x, y, z].
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.
>>> 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
>>> v0, v1 = random_vector(3), random_vector(3)
>>> M = rotation_matrix(angle_between_vectors(v0, v1), vector_product(v0, v1))
>>> v2 = numpy.dot(v0, M[:3,:3].T)
>>> numpy.allclose(unit_vector(v1), unit_vector(v2))
True
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=[1, 0, 0, 0])
>>> 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()
Set initial cursor window coordinates and pick constrain-axis.
Update current cursor window coordinates.
Return state of constrain to axis mode.
Return homogeneous rotation matrix.
Continue rotation in direction of last drag.
Place Arcball, e.g. when window size changes.
Set axes to constrain rotations.
Set state of constrain to axis mode.
Return affine transform matrix to register two point sets.
v0 and v1 are shape (ndims, *) arrays of at least ndims non-homogeneous coordinates, where ndims is the dimensionality of the coordinate space.
If shear is False, a similarity transformation matrix is returned. If also scale is False, a rigid/Eucledian transformation matrix is returned.
By default the algorithm by Hartley and Zissermann [15] is used. If usesvd is True, similarity and Eucledian transformation matrices are calculated by minimizing the weighted sum of squared deviations (RMSD) according to the algorithm by Kabsch [8]. Otherwise, and if ndims is 3, the quaternion based algorithm by Horn [9] is used, which is slower when using this Python implementation.
The returned matrix performs rotation, translation and uniform scaling (if specified).
>>> v0 = [[0, 1031, 1031, 0], [0, 0, 1600, 1600]]
>>> v1 = [[675, 826, 826, 677], [55, 52, 281, 277]]
>>> affine_matrix_from_points(v0, v1)
array([[ 0.14549, 0.00062, 675.50008],
[ 0.00048, 0.14094, 53.24971],
[ 0. , 0. , 1. ]])
>>> T = translation_matrix(numpy.random.random(3)-0.5)
>>> R = random_rotation_matrix(numpy.random.random(3))
>>> S = scale_matrix(random.random())
>>> M = concatenate_matrices(T, R, S)
>>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = numpy.dot(M, v0)
>>> v0[:3] += numpy.random.normal(0, 1e-8, 300).reshape(3, -1)
>>> M = affine_matrix_from_points(v0[:3], v1[:3])
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
More examples in superimposition_matrix()
Return angle between vectors.
If directed is False, the input vectors are interpreted as undirected axes, i.e. the maximum angle is pi/2.
>>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3])
>>> numpy.allclose(a, math.pi)
True
>>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3], directed=False)
>>> numpy.allclose(a, 0)
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> a = angle_between_vectors(v0, v1)
>>> numpy.allclose(a, [0, 1.5708, 1.5708, 0.95532])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> a = angle_between_vectors(v0, v1, axis=1)
>>> numpy.allclose(a, [1.5708, 1.5708, 1.5708, 0.95532])
True
Return sphere point perpendicular to axis.
Return unit sphere coordinates from window coordinates.
Return axis, which arc is nearest to point.
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 (divided by w coordinate).
>>> frustrum = numpy.random.rand(6)
>>> frustrum[1] += frustrum[0]
>>> frustrum[3] += frustrum[2]
>>> frustrum[5] += frustrum[4]
>>> M = clip_matrix(perspective=False, *frustrum)
>>> numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1])
array([-1., -1., -1., 1.])
>>> numpy.dot(M, [frustrum[1], frustrum[3], frustrum[5], 1])
array([ 1., 1., 1., 1.])
>>> M = clip_matrix(perspective=True, *frustrum)
>>> v = numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1])
>>> v / v[3]
array([-1., -1., -1., 1.])
>>> v = numpy.dot(M, [frustrum[1], frustrum[3], frustrum[4], 1])
>>> v / v[3]
array([ 1., 1., -1., 1.])
Return transformation matrix from sequence of transformations.
This is the inverse of the decompose_matrix function.
>>> 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
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
Return sequence of transformations from transformation 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
Distance between two points in space
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*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")
Return Euler angles from quaternion for specified axis sequence.
>>> angles = euler_from_quaternion([0.99810947, 0.06146124, 0, 0])
>>> numpy.allclose(angles, [0.123, 0, 0])
True
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*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)
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))
True
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)
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
Return the x vector normalized
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
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
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
>>> v0[3] = 1
>>> v1 = numpy.dot(P, v0)
>>> numpy.allclose(v1[1], v0[1])
True
>>> numpy.allclose(v1[0], 3-v1[1])
True
Return quaternion for rotation about axis.
>>> q = quaternion_about_axis(0.123, [1, 0, 0])
>>> numpy.allclose(q, [0.99810947, 0.06146124, 0, 0])
True
Return conjugate of quaternion.
>>> q0 = random_quaternion()
>>> q1 = quaternion_conjugate(q0)
>>> q1[0] == q0[0] and all(q1[1:] == -q0[1:])
True
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, [0.435953, 0.310622, -0.718287, 0.444435])
True
Return quaternion from rotation matrix.
If isprecise is True, the input matrix is assumed to be a precise rotation matrix and a faster algorithm is used.
>>> q = quaternion_from_matrix(numpy.identity(4), True)
>>> numpy.allclose(q, [1, 0, 0, 0])
True
>>> q = quaternion_from_matrix(numpy.diag([1, -1, -1, 1]))
>>> numpy.allclose(q, [0, 1, 0, 0]) or numpy.allclose(q, [0, -1, 0, 0])
True
>>> R = rotation_matrix(0.123, (1, 2, 3))
>>> q = quaternion_from_matrix(R, True)
>>> numpy.allclose(q, [0.9981095, 0.0164262, 0.0328524, 0.0492786])
True
>>> R = [[-0.545, 0.797, 0.260, 0], [0.733, 0.603, -0.313, 0],
... [-0.407, 0.021, -0.913, 0], [0, 0, 0, 1]]
>>> q = quaternion_from_matrix(R)
>>> numpy.allclose(q, [0.19069, 0.43736, 0.87485, -0.083611])
True
>>> R = [[0.395, 0.362, 0.843, 0], [-0.626, 0.796, -0.056, 0],
... [-0.677, -0.498, 0.529, 0], [0, 0, 0, 1]]
>>> q = quaternion_from_matrix(R)
>>> numpy.allclose(q, [0.82336615, -0.13610694, 0.46344705, -0.29792603])
True
>>> R = random_rotation_matrix()
>>> q = quaternion_from_matrix(R)
>>> is_same_transform(R, quaternion_matrix(q))
True
Return imaginary part of quaternion.
>>> quaternion_imag([3, 0, 1, 2])
array([ 0., 1., 2.])
Return inverse of quaternion.
>>> q0 = random_quaternion()
>>> q1 = quaternion_inverse(q0)
>>> numpy.allclose(quaternion_multiply(q0, q1), [1, 0, 0, 0])
True
Return homogeneous rotation matrix from quaternion.
>>> M = quaternion_matrix([0.99810947, 0.06146124, 0, 0])
>>> numpy.allclose(M, rotation_matrix(0.123, [1, 0, 0]))
True
>>> M = quaternion_matrix([1, 0, 0, 0])
>>> numpy.allclose(M, numpy.identity(4))
True
>>> M = quaternion_matrix([0, 1, 0, 0])
>>> numpy.allclose(M, numpy.diag([1, -1, -1, 1]))
True
Return multiplication of two quaternions.
>>> q = quaternion_multiply([4, 1, -2, 3], [8, -5, 6, 7])
>>> numpy.allclose(q, [28, -44, -14, 48])
True
Return real part of quaternion.
>>> quaternion_real([3, 0, 1, 2])
3.0
Return spherical linear interpolation between two quaternions.
>>> q0 = random_quaternion()
>>> q1 = random_quaternion()
>>> q = quaternion_slerp(q0, q1, 0)
>>> numpy.allclose(q, q0)
True
>>> q = quaternion_slerp(q0, q1, 1, 1)
>>> numpy.allclose(q, q1)
True
>>> q = quaternion_slerp(q0, q1, 0.5)
>>> angle = math.acos(numpy.dot(q0, q))
>>> numpy.allclose(2, math.acos(numpy.dot(q0, q1)) / angle) or numpy.allclose(2, math.acos(-numpy.dot(q0, q1)) / angle)
True
Return uniform random unit quaternion.
>>> q = random_quaternion()
>>> numpy.allclose(1, vector_norm(q))
True
>>> q = random_quaternion(numpy.random.random(3))
>>> len(q.shape), q.shape[0]==4
(1, True)
Return uniform random rotation matrix.
>>> R = random_rotation_matrix()
>>> numpy.allclose(numpy.dot(R.T, R), numpy.identity(4))
True
Return array of random doubles in the half-open interval [0.0, 1.0).
>>> v = random_vector(10000)
>>> numpy.all(v >= 0) and numpy.all(v < 1)
True
>>> v0 = random_vector(10)
>>> v1 = random_vector(10)
>>> numpy.any(v0 == v1)
False
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
Return matrix to mirror at plane defined by point and normal vector.
>>> v0 = numpy.random.random(4) - 0.5
>>> v0[3] = 1.
>>> 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
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
Create a rotation matrix corresponding to the rotation around a general axis by a specified angle.
R = dd^T + cos(a) (I - dd^T) + sin(a) skew(d)
Parameters:
angle : float a direction : array d
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
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
>>> v[3] = 1
>>> 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)
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
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, numpy.linalg.det(S))
True
Given the parameters for a frustum returns a 4x4 perspective projection matrix
Return: a 4x4 perspective matrix
Return matrix to transform given 3D point set into second point set.
v0 and v1 are shape (3, *) or (4, *) arrays of at least 3 points.
The parameters scale and usesvd are explained in the more general affine_matrix_from_points function.
The returned matrix is a similarity or Eucledian transformation matrix. This function has a fast C implementation in transformations.c.
>>> 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
>>> v0[3] = 1
>>> 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, 1e-9, 300).reshape(3, -1)
>>> M = superimposition_matrix(v0, v1, scale=True)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v0))
True
>>> v = numpy.empty((4, 100, 3))
>>> v[:, :, 0] = v0
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0]))
True
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
Return matrix to translate by direction vector.
>>> v = numpy.random.random(3) - 0.5
>>> numpy.allclose(v, translation_matrix(v)[:3, 3])
True
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))
>>> unit_vector(v0, axis=1, out=v1)
>>> numpy.allclose(v1, v2)
True
>>> list(unit_vector([]))
[]
>>> list(unit_vector([1]))
[1.0]
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))
>>> 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])
1.0
Return vector perpendicular to vectors.
>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> numpy.allclose(v, [0, 0, 6])
True
>>> v0 = [[2, 0, 0, 2], [0, 2, 0, 2], [0, 0, 2, 2]]
>>> v1 = [[3], [0], [0]]
>>> v = vector_product(v0, v1)
>>> numpy.allclose(v, [[0, 0, 0, 0], [0, 0, 6, 6], [0, -6, 0, -6]])
True
>>> v0 = [[2, 0, 0], [2, 0, 0], [0, 2, 0], [2, 0, 0]]
>>> v1 = [[0, 3, 0], [0, 0, 3], [0, 0, 3], [3, 3, 3]]
>>> v = vector_product(v0, v1, axis=1)
>>> numpy.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True