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c}} Z=dwdaZ>dwdbZ?dvd+e fdcZ@dddZAddeZBdf ZCdg ZDdh ZEdi ZFdj ZGddkZHdl ZId}dmZJ ddnedoed/eKdpeej fdqZMddrZNddsZOdwdtZPdu ZQyc c}} w )a$ 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.
:Author:
`Christoph Gohlke `_
:Organization:
Laboratory for Fluorescence Dynamics, University of California, Irvine
:Version: 2017.02.17
Requirements
------------
* `CPython 2.7 or 3.4 `_
* `numpy 1.9 `_
* `Transformations.c 2015.03.19 `_
(recommended for speedup of some functions)
Notes
-----
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 np.linalg.inv(M), be concatenated using
np.dot(M0, M1), or transform homogeneous coordinate arrays (v) using
np.dot(M, v) for shape (4, *) column vectors, respectively
np.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 np.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.
Other Python packages and modules for 3D transformations and quaternions:
* `Transforms3d `_
includes most code of this module.
* `Blender.mathutils `_
* `numpy-dtypes `_
References
----------
(1) Matrices and transformations. Ronald Goldman.
In "Graphics Gems I", pp 472-475. Morgan Kaufmann, 1990.
(2) More matrices and transformations: shear and pseudo-perspective.
Ronald Goldman. In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991.
(3) Decomposing a matrix into simple transformations. Spencer Thomas.
In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991.
(4) Recovering the data from the transformation matrix. Ronald Goldman.
In "Graphics Gems II", pp 324-331. Morgan Kaufmann, 1991.
(5) Euler angle conversion. Ken Shoemake.
In "Graphics Gems IV", pp 222-229. Morgan Kaufmann, 1994.
(6) Arcball rotation control. Ken Shoemake.
In "Graphics Gems IV", pp 175-192. Morgan Kaufmann, 1994.
(7) Representing attitude: Euler angles, unit quaternions, and rotation
vectors. James Diebel. 2006.
(8) A discussion of the solution for the best rotation to relate two sets
of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828.
(9) Closed-form solution of absolute orientation using unit quaternions.
BKP Horn. J Opt Soc Am A. 1987. 4(4):629-642.
(10) Quaternions. Ken Shoemake.
http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf
(11) From quaternion to matrix and back. JMP van Waveren. 2005.
http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm
(12) Uniform random rotations. Ken Shoemake.
In "Graphics Gems III", pp 124-132. Morgan Kaufmann, 1992.
(13) Quaternion in molecular modeling. CFF Karney.
J Mol Graph Mod, 25(5):595-604
(14) New method for extracting the quaternion from a rotation matrix.
Itzhack Y Bar-Itzhack, J Guid Contr Dynam. 2000. 23(6): 1085-1087.
(15) Multiple View Geometry in Computer Vision. Hartley and Zissermann.
Cambridge University Press; 2nd Ed. 2004. Chapter 4, Algorithm 4.7, p 130.
(16) Column Vectors vs. Row Vectors.
http://steve.hollasch.net/cgindex/math/matrix/column-vec.html
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')
>>> np.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(np.random.rand(3))
>>> M = concatenate_matrices(T, R, Z, S)
>>> scale, shear, angles, trans, persp = decompose_matrix(M)
>>> np.allclose(scale, 1.23)
True
>>> np.allclose(trans, [1, 2, 3])
True
>>> np.allclose(shear, [0, np.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 = np.dot(v0, M[:3,:3].T)
>>> np.allclose(unit_vector(v1), unit_vector(v2))
True
N) ArrayLikeNDArray )IntegerNumberSeed) diagonal_dotrandom_generator F WRITEABLEc , t j d S )zReturn 4x4 identity/unit matrix.
>>> I = identity_matrix()
>>> np.allclose(I, np.dot(I, I))
True
>>> float(np.sum(I)), float(np.trace(I))
(4.0, 4.0)
>>> np.allclose(I, np.identity(4))
True
r )npidentity h/workspace/moltex-experiments/.pixi/envs/default/lib/python3.12/site-packages/trimesh/transformations.pyidentity_matrixr s ;;q>r c t | }t d | D rddlm} ||dz }nt j |dz }| d| |d||f< |S )z
Return matrix to translate by direction vector.
>>> v = np.random.random(3) - 0.5
>>> np.allclose(v, translation_matrix(v)[:3, 3])
True
c 3 H K | ] }d t t | v yw)sympyN)strtype).0vs r z%translation_matrix.. s
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66 aL FF37O Tc?AdsdCiL Hr c V t j | dddf j S )zReturn translation vector from translation matrix.
>>> v0 = np.random.random(3) - 0.5
>>> v1 = translation_from_matrix(translation_matrix(v0))
>>> np.allclose(v0, v1)
True
N )r asarraycopymatrixs r translation_from_matrixr) s) ::f bqb!e $ ) ) ++r c t |dd }t j d }|ddddfxx dt j || z z cc< dt j | dd | z |z |dddf< |S )a Return matrix to mirror at plane defined by point and normal vector.
>>> v0 = np.random.random(4) - 0.5
>>> v0[3] = 1.
>>> v1 = np.random.random(3) - 0.5
>>> R = reflection_matrix(v0, v1)
>>> np.allclose(2, np.trace(R))
True
>>> np.allclose(v0, np.dot(R, v0))
True
>>> v2 = v0.copy()
>>> v2[:3] += v1
>>> v3 = v0.copy()
>>> v2[:3] -= v1
>>> np.allclose(v2, np.dot(R, v3))
True
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dk d }t | s t d t j |dd|d f j }||d z }||fS )
aK Return mirror plane point and normal vector from reflection matrix.
>>> v0 = np.random.random(3) - 0.5
>>> v1 = np.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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Return matrix to rotate about axis defined by point and
direction.
Parameters
-------------
angle : float, or sympy.Symbol
Angle, in radians or symbolic angle
direction : (3,) float
Any vector along rotation axis
point : (3, ) float, or None
Origin point of rotation axis
Returns
-------------
matrix : (4, 4) float, or (4, 4) sympy.Matrix
Homogeneous transformation matrix
Examples
-------------
>>> R = rotation_matrix(np.pi/2, [0, 0, 1], [1, 0, 0])
>>> np.allclose(np.dot(R, [0, 0, 0, 1]), [1, -1, 0, 1])
True
>>> angle = (random.random() - 0.5) * (2*np.pi)
>>> direc = np.random.random(3) - 0.5
>>> point = np.random.random(3) - 0.5
>>> R0 = rotation_matrix(angle, direc, point)
>>> R1 = rotation_matrix(angle-2*np.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 = np.identity(4, np.float64)
>>> np.allclose(I, rotation_matrix(np.pi*2, direc))
True
>>> np.allclose(2, np.trace(rotation_matrix(np.pi/2,direc,point)))
True
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| } | ||fS )a Return rotation angle and axis from rotation matrix.
>>> angle = (random.random() - 0.5) * (2*np.pi)
>>> direc = np.random.random(3) - 0.5
>>> point = np.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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9Q<4 $4#:1 5 ! DD RS T Yq\ T !$4#:1 5 ! DD RS T$4#:1 5 ! DD RS T JJtT "E )U ""r c x |=t j | | | dg }| |dd |dddf< |dddfxx d| z
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|dd | z |z |dddf< |S )a Return matrix to scale by factor around origin in direction.
Use factor -1 for point symmetry.
>>> v = (np.random.rand(4, 5) - 0.5) * 20
>>> v[3] = 1
>>> S = scale_matrix(-1.234)
>>> np.allclose(np.dot(S, v)[:3], -1.234*v[:3])
True
>>> factor = random.random() * 10 - 5
>>> origin = np.random.random(3) - 0.5
>>> direct = np.random.random(3) - 0.5
>>> S = scale_matrix(factor, origin)
>>> S = scale_matrix(factor, origin, direct)
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dk d }t | s t d t j |dd|d
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|dz dz }d}Y w xY w) aW Return scaling factor, origin and direction from scaling matrix.
>>> factor = random.random() * 10 - 5
>>> origin = np.random.random(3) - 0.5
>>> direct = np.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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| | |z |dddf< |S ) ae 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])
>>> np.allclose(P[1:, 1:], np.identity(4)[1:, 1:])
True
>>> point = np.random.random(3) - 0.5
>>> normal = np.random.random(3) - 0.5
>>> direct = np.random.random(3) - 0.5
>>> persp = np.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, np.dot(P0, P3))
True
>>> P = projection_matrix([3, 0, 0], [1, 1, 0], [1, 0, 0])
>>> v0 = (np.random.rand(4, 5) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = np.dot(P, v0)
>>> np.allclose(v1[1], v0[1])
True
>>> np.allclose(v1[0], 3-v1[1])
True
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|fS ) a 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 = np.random.random(3) - 0.5
>>> normal = np.random.random(3) - 0.5
>>> direct = np.random.random(3) - 0.5
>>> persp = np.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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z gg dg}t j | S )aC Return matrix to obtain normalized device coordinates from frustum.
The frustum 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 frustum.
If perspective is True the frustum 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).
>>> frustum = np.random.rand(6)
>>> frustum[1] += frustum[0]
>>> frustum[3] += frustum[2]
>>> frustum[5] += frustum[4]
>>> M = clip_matrix(perspective=False, *frustum)
>>> a = np.dot(M, [frustum[0], frustum[2], frustum[4], 1])
>>> np.allclose(a, [-1., -1., -1., 1.])
True
>>> b = np.dot(M, [frustum[1], frustum[3], frustum[5], 1])
>>> np.allclose(b, [ 1., 1., 1., 1.])
True
>>> M = clip_matrix(perspective=True, *frustum)
>>> v = np.dot(M, [frustum[0], frustum[2], frustum[4], 1])
>>> c = v / v[3]
>>> np.allclose(c, [-1., -1., -1., 1.])
True
>>> v = np.dot(M, [frustum[1], frustum[3], frustum[4], 1])
>>> d = v / v[3]
>>> np.allclose(d, [ 1., 1., -1., 1.])
True
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cc< | t j |dd | z |z |dddf< |S )a 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*np.pi
>>> direct = np.random.random(3) - 0.5
>>> point = np.random.random(3) - 0.5
>>> normal = np.cross(direct, np.random.random(3))
>>> S = shear_matrix(angle, direct, point, normal)
>>> np.allclose(1, np.linalg.det(S))
True
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fS )a Return shear angle, direction and plane from shear matrix.
>>> angle = np.pi / 2.0
>>> direct = [0.0, 1.0, 0.0]
>>> point = [0.0, 0.0, 0.0]
>>> normal = np.cross(direct, np.roll(direct,1))
>>> 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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< |||||fS )a 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)
>>> np.allclose(T0, T1)
True
>>> S = scale_matrix(0.123)
>>> scale, shear, angles, trans, persp = decompose_matrix(S)
>>> bool(np.isclose(scale[0], 0.123))
True
>>> R0 = euler_matrix(1, 2, 3)
>>> scale, shear, angles, trans, persp = decompose_matrix(R0)
>>> R1 = euler_matrix(*angles)
>>> np.allclose(R0, R1)
True
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}||d z }|S )aR 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 = np.random.random(3) - 0.5
>>> shear = np.random.random(3) - 0.5
>>> angles = (np.random.random(3) - 0.5) * (2*np.pi)
>>> trans = np.random.random(3) - 0.5
>>> persp = np.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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Angles are expected in degrees.
The de-orthogonalization matrix is the inverse.
>>> O = orthogonalization_matrix([10, 10, 10], [90, 90, 90])
>>> np.allclose(O[:3, :3], np.identity(3, float) * 10)
True
>>> O = orthogonalization_matrix([9.8, 12.0, 15.5], [87.2, 80.7, 69.7])
>>> np.allclose(np.sum(O), 43.063229)
True
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gg} t j j) | \ } } | ddt j* | f } | t- | z } t/ | }|r[|sY| | z } ||z }|d|d|fxx t j0 t j$ | t j$ | z z cc< t j t j j3 | t j || }||||f z }|S )a 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/Euclidean transformation matrix
is returned.
By default the algorithm by Hartley and Zissermann [15] is used.
If usesvd is True, similarity and Euclidean 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]]
>>> mat = affine_matrix_from_points(v0, v1)
>>> T = translation_matrix(np.random.random(3)-0.5)
>>> R = random_rotation_matrix(np.random.random(3))
>>> S = scale_matrix(random.random())
>>> M = concatenate_matrices(T, R, S)
>>> v0 = (np.random.rand(4, 100) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = np.dot(M, v0)
>>> v0[:3] += np.random.normal(0, 1e-8, 300).reshape(3, -1)
>>> M = affine_matrix_from_points(v0[:3], v1[:3])
>>> check = np.allclose(v1, np.dot(M, v0))
More examples in superimposition_matrix()
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b &5&&5&.RWWRVVBZ"&&*%< == ryy}}R "&&B-0A5%< A Hr c t j | t j dd } t j |t j dd }t | |d|| S )a# 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 Euclidean transformation matrix.
This function has a fast C implementation in transformations.c.
>>> v0 = np.random.rand(3, 10)
>>> M = superimposition_matrix(v0, v0)
>>> np.allclose(M, np.identity(4))
True
>>> R = random_rotation_matrix(np.random.random(3))
>>> v0 = [[1,0,0], [0,1,0], [0,0,1], [1,1,1]]
>>> v1 = np.dot(R, v0)
>>> M = superimposition_matrix(v0, v1)
>>> np.allclose(v1, np.dot(M, v0))
True
>>> v0 = (np.random.rand(4, 100) - 0.5) * 20
>>> v0[3] = 1
>>> v1 = np.dot(R, v0)
>>> M = superimposition_matrix(v0, v1)
>>> np.allclose(v1, np.dot(M, v0))
True
>>> S = scale_matrix(random.random())
>>> T = translation_matrix(np.random.random(3)-0.5)
>>> M = concatenate_matrices(T, R, S)
>>> v1 = np.dot(M, v0)
>>> v0[:3] += np.random.normal(0, 1e-9, 300).reshape(3, -1)
>>> M = superimposition_matrix(v0, v1, scale=True)
>>> np.allclose(v1, np.dot(M, v0))
True
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> np.allclose(v1, np.dot(M, v0))
True
>>> v = np.zeros((4, 100, 3))
>>> v[:, :, 0] = v0
>>> M = superimposition_matrix(v0, v1, scale=True, usesvd=False)
>>> np.allclose(v1, np.dot(M, v[:, :, 0]))
True
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f< |S # t t f$ r t | |\ }}}}Y w xY w)av 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')
>>> np.allclose(np.sum(R[0]), -1.34786452)
True
>>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1))
>>> np.allclose(np.sum(R[0]), -0.383436184)
True
>>> ai, aj, ak = (4*np.pi) * (np.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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fS # t t f$ r t | |\ }}}}Y 4w xY w)a 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')
>>> np.allclose(R0, R1)
True
>>> angles = (4*np.pi) * (np.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 np.allclose(R0, R1): print(axes, "failed")
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H % 4D/3, 6:u4s H0 0 IIc , t t | | S )zReturn Euler angles from quaternion for specified axis sequence.
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>>> np.allclose(angles, [0.123, 0, 0])
True
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ai, aj, ak : Euler's roll, pitch and yaw angles
axes : One of 24 axis sequences as string or encoded tuple
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True
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| dz |d< |S )zReturn quaternion for rotation about axis.
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True
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Return a homogeneous rotation matrix from quaternion.
>>> M = quaternion_matrix([0.99810947, 0.06146124, 0, 0])
>>> np.allclose(M, rotation_matrix(0.123, [1, 0, 0]))
True
>>> M = quaternion_matrix([1, 0, 0, 0])
>>> np.allclose(M, np.identity(4))
True
>>> M = quaternion_matrix([0, 1, 0, 0])
>>> np.allclose(M, np.diag([1, -1, -1, 1]))
True
>>> M = quaternion_matrix([[1, 0, 0, 0],[0, 1, 0, 0]])
>>> np.allclose(M, np.array([np.identity(4), np.diag([1, -1, -1, 1])]))
True
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|| z |z gg }|dz }t j j | \ }}|g dt j | f }|d dk rt j || |S )a 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(np.identity(4), True)
>>> np.allclose(q, [1, 0, 0, 0])
True
>>> q = quaternion_from_matrix(np.diag([1, -1, -1, 1]))
>>> np.allclose(q, [0, 1, 0, 0]) or np.allclose(q, [0, -1, 0, 0])
True
>>> R = rotation_matrix(0.123, (1, 2, 3))
>>> q = quaternion_from_matrix(R, True)
>>> np.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)
>>> np.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)
>>> np.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
>>> is_same_quaternion(quaternion_from_matrix(R, isprecise=False),
... quaternion_from_matrix(R, isprecise=True))
True
>>> R = euler_matrix(0.0, 0.0, np.pi/2.0)
>>> is_same_quaternion(quaternion_from_matrix(R, isprecise=False),
... quaternion_from_matrix(R, isprecise=True))
True
r3 Nr r rn r rT r r$ rU )rG r rG rV r r )r r rG rl rk )r rG r rm )rG r r )r$ r r rG ?rF rd ) r r% r9 r rX r rK r: r r r )r( ispreciser! r r rC r r m00m01m02m10m11m12m20m21m22KrA rB s r quaternion_from_matrixr Z s N
6,RaR!V4A HHTN HHQK qw;AaDT7QtW$AaDT7QtW$AaDT7QtW$AaD GAq!w4 !1aw1a4 !1a!Q$1QT7Qq!tW,-$7AAaDQT7Qq!tW$AaDQT7Qq!tW$AaDQT7Qq!tW$AaD,A S2771qw;'
''ggggggggg HHsS #sC0sC#IOS#6sC#IsSy3<sC#IsSy#)c/B
Syy~~a 1
lBIIaL( )tcz
Aq Hr c |\ }}}}| \ }}}} t j | |z ||z z
| |z z
||z z ||z ||z z | |z z
||z z | |z ||z z | |z z ||z z ||z ||z z
| |z z ||z z gt j S )zReturn multiplication of two quaternions.
>>> q = quaternion_multiply([4, 1, -2, 3], [8, -5, 6, 7])
>>> np.allclose(q, [28, -44, -14, 48])
True
r3 r rK r9 )
quaternion1 quaternion0w0x0y0z0w1x1y1z1s
r quaternion_multiplyr# s !NBB NBB
88
C"HrBw b (27 2 Gb2g R '"r' 1
C"HrBw b (27 2 Gb2g R '"r' 1
jj r c t j | t j }t j |dd |dd |S )zReturn conjugate of quaternion.
>>> q0 = random_quaternion()
>>> q1 = quaternion_conjugate(q0)
>>> q1[0] == q0[0] and all(q1[1:] == -q0[1:])
True
r3 r N)r rK r9 r r r s r quaternion_conjugater& s7 2::.AKK!"qu Hr c t j | t j }t j |dd |dd |t j || z S )zReturn inverse of quaternion.
>>> q0 = random_quaternion()
>>> q1 = quaternion_inverse(q0)
>>> np.allclose(quaternion_multiply(q0, q1), [1, 0, 0, 0])
True
r3 r N)r rK r9 r r. r% s r quaternion_inverser( sG 2::.AKK!"qu rvva| r c t | d S )zTReturn real part of quaternion.
>>> quaternion_real([3, 0, 1, 2])
3.0
r )floatr s r quaternion_realr, s A r c R t j | dd t j S )ziReturn imaginary part of quaternion.
>>> quaternion_imag([3, 0, 1, 2])
array([0., 1., 2.])
r r r3 r r+ s r quaternion_imagr. s 88JqO2:: 66r c 8 t | dd }t |dd }|dk( r|S |dk( r|S t j || }t t | dz
t k r|S |r |dk r| }t j
|| t j | |t j z z }t | t k r|S dt j | z } |t j d|z
|z | z z }|t j ||z | z z }||z
}|S )a Return spherical linear interpolation between two quaternions.
>>> q0 = random_quaternion()
>>> q1 = random_quaternion()
>>> q = quaternion_slerp(q0, q1, 0)
>>> np.allclose(q, q0)
True
>>> q = quaternion_slerp(q0, q1, 1, 1)
>>> np.allclose(q, q1)
True
>>> q = quaternion_slerp(q0, q1, 0.5)
>>> angle = np.arccos(np.dot(q0, q))
>>> np.allclose(2, np.arccos(np.dot(q0, q1)) / angle) or np.allclose(2, np.arccos(-np.dot(q0, q1)) / angle)
True
Nr rF r5 ) r, r r. r= rx r arccospirH )
quat0quat1fractionspin shortestpathq0q1drM isins
r quaternion_slerpr; s $
U2AY B U2AY B3 S
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Ir seedc F | -t | j d|z j d } n| j d dk( sJ t j
d| d z
}t j
| d }t j dz }|| d z }|| d z }t j t j | |z t j | |z t j | |z t j | |z g j j S )al Return uniform random unit quaternion.
rand: array like or None
Three independent random variables that are uniformly distributed
between 0 and 1.
seed : None or int
Seed for deterministic results, otherwise OS entropy.
>>> q = random_quaternion()
>>> np.allclose(1, vector_norm(q))
True
>>> q = random_quaternion(num=10)
>>> np.allclose(1, vector_norm(q, axis=1))
True
>>> q = random_quaternion(random_vector(3))
>>> len(q.shape), q.shape[0]==4
(1, True)
>>> bool(np.allclose(random_quaternion(seed=0), random_quaternion(seed=0)))
True
r$ )r$ r8 r r5 r+ r rG ) r
randomr r r r r1 rK rI rH rW r@ )randnumr< r1r2pi2r t2s r random_quaternionrE ! s , | %,,QW5==gFzz!}!!! tAw B a B
%%#+C tAw B tAw B
88 b "&&*r/266":?BFF2JOL a r r? r@ r c t | }t t | || }|r'|j d dz
t | z |dddf< |S )a\
Return uniform random rotation matrix.
Parameters
------------
rand : (3,)
Three independent random variables that are uniformly distributed
between 0 and 1 for each returned quaternion.
num
Number of matrices to return.
translate
If passed the rotation matrix will include translation
that is random and between positive and negative half this value.
seed
Seed for deterministic results, otherwise OS entropy.
>>> R = random_rotation_matrix()
>>> np.allclose(np.dot(R.T, R), np.identity(4))
True
>>> R = random_rotation_matrix(num=10)
>>> np.allclose(np.einsum('...ji,...jk->...ik', R, R), np.identity(4))
True
>>> bool(np.allclose(random_rotation_matrix(seed=0, translate=10),
... random_rotation_matrix(seed=0, translate=10)))
True
)r? r@ r< r$ r
N)r
r rE r> r* )r? r@ r r< r> r( s r random_rotation_matrixrG E sT F d
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RF q)C/53CCrr1u
Mr c r e Zd ZdZd dZd Zd Zed Zej d Zd Z
d Z d
d
Z d Z
y)ArcballaO Virtual Trackball Control.
>>> ball = Arcball()
>>> ball = Arcball(initial=np.identity(4))
>>> ball.place([320, 320], 320)
>>> ball.down([500, 250])
>>> ball.drag([475, 275])
>>> R = ball.matrix()
>>> np.allclose(np.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.constrain = True
>>> ball.down([400, 200])
>>> ball.drag([200, 400])
>>> R = ball.matrix()
>>> np.allclose(np.sum(R), 0.2055924)
True
>>> ball.next()
Nc d| _ d| _ d| _ ddg| _ t j
g d | _ d| _ | t j
g d | _ nut j
|t j }|j dk( rt | | _ n0|j d k( r|t | z }|| _ n t d
| j x| _ | _ y) z`Initialize virtual trackball control.
initial : quaternion or rotation matrix
Nr5 rF )rF rF r5 F)r5 rF rF rF r3 r r r z"initial not a quaternion or matrix)_axis_axes_radius_centerr rK _vdown
_constrain_qdownr9 r r rf r? _qnow_qpre)selfinitials r __init__zArcball.__init__ s
Sz hh/ ? ((#78DKhhwbjj9G}}& 4W = $&;w// % !EFF"&++-
TZr c l t | | _ |d | j d< |d | j d< y)zPlace Arcball, e.g. when window size changes.
center : sequence[2]
Window coordinates of trackball center.
radius : float
Radius of trackball in window coordinates.
r r N)r* rN rO )rU centerradiuss r placez
Arcball.place s1 V} ) Q ) Qr c ^ |d| _ y|D cg c]
}t | c}| _ yc c}w )z Set axes to constrain rotations.N)rM r, )rU r r s r setaxeszArcball.setaxes s, < DJ8<=+d+=DJ=s *c | j S )z'Return state of constrain to axis mode.)rQ rU s r constrainzArcball.constrain s r c $ t | | _ y)z$Set state of constrain to axis mode.N)boolrQ )rU values r r` zArcball.constrain s u+r c T t || j | j | _ | j x| _ | _ | j rW| j Kt | j | j | _
t | j | j | _ yd| _
y)z>Set initial cursor window coordinates and pick constrain-axis.N) arcball_map_to_sphererO rN rP rS rR rT rQ rM arcball_nearest_axisrL arcball_constrain_to_axis)rU r/ s r downz Arcball.down so +E4<< N #'::- dj ??tzz 5-dkk4::FDJ3DKKLDK DJr c t || j | j }| j t || j }| j
| _ t j | j | }t j || t k r| j | _ yt j | j | |d |d |d g}t || j | _ y)z)Update current cursor window coordinates.Nr r rG )re rO rN rL rg rS rT r r rP r. rx rR r# )rU r/ vnowr r s r dragz Arcball.drag s $UDLL$,,G :: !,T4::>DZZ
HHT[[$ '
66!Q<$ DJ T*AaD!A$!=A,Q & & =/
-r rI c | d |d z
|z }|d | d z
|z }||z ||z z }|dkD r3t j | }t j ||z ||z dg S t j ||t j d|z
g S )z7Return unit sphere coordinates from window coordinates.r r r5 rF )r r rK )r/ rY rZ r r r s r re re s
(VAY
& (B
)eAh
& (B
R"r'A3w GGAJxxaa-..xxRq!1233r c t j | t j }t j |t j }||t j || z z }t | }|t
kD r%|d dk rt j || ||z }|S |d dk( rt j g d S t |d |d dg S )z*Return sphere point perpendicular to axis.r3 rG rF r5 )r5 rF rF r r )r rK r9 r. rf rx r r, )r/ r r r r s r rg rg s
bjj)A
RZZ(ARVVAq\ AAA4x Q4#: KK1 Qts{xx(( 1qtS) **r c t j | t j } d}d}|D ], }t j t | | | }||kD s)|}|}. |S )z+Return axis, which arc is nearest to point.r3 Nrv )r r% r9 r. rg )r/ r nearestmxr r s r rf rf sZ JJuBJJ /EG
B FF,UD95 A r6GB
Nr g @)r rG r r )r r r r sxyx)r r r r sxzy)r r r r sxzx)r r r r syzx)r r r r syzy)r r r r syxz)r r r r syxy)r r r r szxy)rG r r r szxz)rG r r r szyx)rG r r r szyz)rG r r r rzyxr r r r rxyx)r r r r ryzx)r r r r rxzx)r r r r rxzy)r r r r )r r r r )r r r r )r r r r )rG r r r )rG r r r )rG r r r )rG r r r )ryzyrzxyryxyryxzrzxzrxyzrzyzc t j | t j } || j dk( r)t j t j
| | S | | z } t j t j | | }t j || |S | | z } t j | || t j || y)a Return length, i.e. Euclidean norm, of ndarray along axis.
>>> v = np.random.random(3)
>>> n = vector_norm(v)
>>> np.allclose(n, np.linalg.norm(v))
True
>>> v = np.random.rand(6, 5, 3)
>>> n = vector_norm(v, axis=-1)
>>> np.allclose(n, np.sqrt(np.sum(v*v, axis=2)))
True
>>> n = vector_norm(v, axis=1)
>>> np.allclose(n, np.sqrt(np.sum(v*v, axis=1)))
True
>>> v = np.random.rand(5, 4, 3)
>>> n = np.zeros((5, 3))
>>> vector_norm(v, axis=1, out=n)
>>> np.allclose(n, np.sqrt(np.sum(v*v, axis=1)))
True
>>> float(vector_norm([]))
0.0
>>> float(vector_norm([1]))
1.0
r3 Nr r )r out)r rK r9 ndimr r.
atleast_1dr )datar r s r rf rf 0 s 2 88D
+D
{ 99>77266$-. . mmBFF4d 34
S
t$C(
Sr c |bt j | t j } | j dk( rL| t j t j
| | z } | S || urt j | |dd |} t j t j | | z | }t j || |t j || }| |z } || S y)a Return ndarray normalized by length, i.e. Euclidean norm, along axis.
>>> v0 = np.random.random(3)
>>> v1 = unit_vector(v0)
>>> np.allclose(v1, v0 / np.linalg.norm(v0))
True
>>> v0 = np.random.rand(5, 4, 3)
>>> v1 = unit_vector(v0, axis=-1)
>>> v2 = v0 / np.expand_dims(np.sqrt(np.sum(v0*v0, axis=2)), 2)
>>> np.allclose(v1, v2)
True
>>> v1 = unit_vector(v0, axis=1)
>>> v2 = v0 / np.expand_dims(np.sqrt(np.sum(v0*v0, axis=1)), 1)
>>> np.allclose(v1, v2)
True
>>> v1 = np.zeros((5, 4, 3))
>>> unit_vector(v0, axis=1, out=v1)
>>> np.allclose(v1, v2)
True
>>> list(unit_vector([]))
[]
>>> [float(i) for i in unit_vector([1])]
[1.0]
Nr3 r )
r rK r9 r r r. r% r r expand_dims)r r r lengths r r, r, W s 4 {xxBJJ/ 99> BGGBFF4 ./ /DK d?ZZ%CF
]]266$+t4
5FGGFF -FND
{ r c 6 t | j | S )ad Return array of random doubles in the half-open interval [0.0, 1.0).
>>> v = random_vector(10000)
>>> bool(np.all(v >= 0) and np.all(v < 1))
True
>>> v0 = random_vector(10)
>>> v1 = random_vector(10)
>>> bool(np.any(v0 == v1))
False
>>> bool(np.allclose(random_vector(10, seed=0), random_vector(10, seed=0)))
True
)r
r> )sizer< s r
random_vectorr s D ! ( ( ..r c 2 t j | || S )a? Return vector perpendicular to vectors.
>>> v = vector_product([2, 0, 0], [0, 3, 0])
>>> np.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)
>>> np.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)
>>> np.allclose(v, [[0, 0, 6], [0, -6, 0], [6, 0, 0], [0, -6, 6]])
True
r )r r )r r r s r vector_productr s $ 88B &&r c t j | t j } t j |t j }t j | |z | }|t | | t || z z }t j
|dd }t j |r| S t j | S )a 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])
>>> np.allclose(a, np.pi)
True
>>> a = angle_between_vectors([1, -2, 3], [-1, 2, -3], directed=False)
>>> np.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)
>>> np.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)
>>> np.allclose(a, [1.5708, 1.5708, 1.5708, 0.95532])
True
r3 r rv r5 )r r% r9 r rf clipr0 fabs)r r directedr r. s r angle_between_vectorsr s 0
Bbjj )B Bbjj )B
&&bt
$C;r % BT(B BBC
''#tS
!C
99HS 77"''#, 77r c @ t j j | S )ah Return inverse of square transformation matrix.
>>> M0 = random_rotation_matrix()
>>> M1 = inverse_matrix(M0.T)
>>> np.allclose(M1, np.linalg.inv(M0.T))
True
>>> for size in range(1, 7):
... M0 = np.random.rand(size, size)
... M1 = inverse_matrix(M0)
... if not np.allclose(M1, np.linalg.inv(M0)): print(size)
)r r: r r' s r inverse_matrixr s 99== r c j t j d }| D ] }t j || } |S )zReturn concatenation of series of transformation matrices.
>>> M = np.random.rand(16).reshape((4, 4)) - 0.5
>>> np.allclose(M, concatenate_matrices(M))
True
>>> np.allclose(np.dot(M, M.T), concatenate_matrices(M, M.T))
True
r )r r r. )matricesr! rC s r concatenate_matricesr s/ AA
FF1aL Hr c t j | t j } | | d z } t j |t j }||d z }t j | | S )zReturn True if two matrices perform same transformation.
>>> is_same_transform(np.identity(4), np.identity(4))
True
>>> is_same_transform(np.identity(4), random_rotation_matrix())
False
r3 rn )r rK r9 allclose)matrix0matrix1s r is_same_transformr sY hhwbjj1G wt} Ghhwbjj1G wt} G
;;w ((r c t j | } t j | }t j | | xs t j | | S )z)Return True if two quaternions are equal.)r rK r )r7 r8 s r is_same_quaternionr s? " B " B
;;r2 6"++b2#"66r c b t j | }t j | } t | }| j |dz |dz fk7 r t d t j
|dz }| |d||f< t j | | }||d||f< t j || }|S )a#
Given a transformation matrix, apply its rotation
around a point in space.
Parameters
----------
matrix: (4,4) or (3, 3) float, transformation matrix
point: (3,) or (2,) float, point in space
Returns
---------
result: (4,4) transformation matrix
r zmatrix must be (d+1, d+1)N)r
asanyarrayr r r? r r. )r( r/ r r results r transform_aroundr s
MM% E
]]6
"F
e*C
||aq))455sQwI !6IdsdCi
VVFI
&F IdsdCi
VVIv
&F Mr c j | ddg} |d}t j | t j } t | }t j | s t d | j dk7 r t d t j dt j }t j | }t j | }||g|ddd f< | |g|d
dd f< | |dd d f< |
t || }|>t j d }|dd dd fxx |z cc< t j || }|S ) au
2D homogeonous transformation matrix.
Parameters
----------
offset : (2,) float
XY offset
theta : float
Rotation around Z in radians
point : (2, ) float
Point to rotate around
scale : (2,) float or None
Scale to apply
Returns
----------
matrix : (3, 3) flat
Homogeneous 2D transformation matrix
NrF r3 ztheta must be finite angle!)rG zoffset must be length 2!r$ r rG r )r( r/ ) r r r9 r* isfiniter? r r rH rI r r. )offsetthetar/ rq rW r r r s r
planar_matrixr s' ( ~s }
]]6
4F %LE
;;u 677
||t344
q
#A
u
A
u
A1vAa!eHAwAa!eHAbqb!eH AU 3 FF1I "1"bqb& U FF1aL Hr c t j | t j } | j dk7 r t d t j
d }| dddf |dddf< | ddddf |ddddf< |S )a+
Given a 2D homogeneous rotation matrix convert it to a 3D rotation
matrix that is rotating around the Z axis
Parameters
----------
matrix_2D: (3,3) float, homogeneous 2D rotation matrix
Returns
----------
matrix_3D: (4,4) float, homogeneous 3D rotation matrix
r3 rn z.Homogeneous 2D transformation matrix required!r NrG r$ )r r r9 r r? r ) matrix_2D matrix_3Ds r planar_matrix_to_3Dr Q s
irzz:I& IJJq I !Q'Ibqb!e!"1"bqb&)Ibqb"1"f r c $ t d|| | }|S )a
Give a spherical coordinate vector, find the rotation that will
transform a [0,0,1] vector to those coordinates
Parameters
-----------
theta: float, rotation angle in radians
phi: float, rotation angle in radians
Returns
----------
matrix: (4,4) rotation matrix where the following will
be a cartesian vector in the direction of the
input spherical coordinates:
np.dot(matrix, [0,0,1,0])
rF )r )r )r phir r s r spherical_matrixr l s $ #sE
5F Mr pointsr( returnc t j | t j } t | dk( s|| j S t j |t j }| j
\ }}t j |t d|dz d|dz f z
j dk r#t j | j S |r]t j | t j | f }t j ||j j ddd|f S t j |d|d|f | j j S )a
Returns points rotated by a homogeneous
transformation matrix.
If points are (n, 2) matrix must be (3, 3)
If points are (n, 3) matrix must be (4, 4)
Parameters
----------
points : (n, dim) float
Points where `dim` is 2 or 3.
matrix : (3, 3) or (4, 4) float
Homogeneous rotation matrix.
translate : bool
Apply translation from matrix or not.
Returns
----------
transformed : (n, dim) float
Transformed points.
r3 r Nr r6 )r r r9 r r& r r= _IDENTITYmaxascontiguousarray column_stackonesr. rW )r r( r countr stacks r transform_pointsr s 0 ]]6
4F
6{a6>{{} ]]6
4F JE3
vvfy37IcAgI!5667;;=D##FKKM22 89vvfegg&((DSD11 66&#tt$fhh / 1 11r c " | j d dz
}t j t j | d|d|f | d|d|f j t
d|d|f z
j
}|dkD r||k rt j j | d|d|f \ }}}t j |dz }t j || |d|d|f< | d||f |d||f< t j || | sJ |S | S )a
If a homogeneous transformation matrix is *almost* a rigid
transform but many matrix-multiplies have accumulated some
floating point error try to restore the matrix using SVD.
Parameters
-----------
matrix : (4, 4) or (3, 3) float
Homogeneous transformation matrix.
max_deviance : float
Do not alter the matrix if it is not rigid by more
than this amount.
Returns
----------
repaired : (4, 4) or (3, 3) float
Repaired homogeneous transformation matrix
r r NgvIh%<=)atol) r r r= r. rW r r r: r r r )r( max_deviancer checkUr rB repaireds r fix_rigidr s# & ,,q/A
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Check to make sure a homogeonous transformation
matrix is a rigid transform.
Parameters
-----------
matrix : (4, 4) float
A transformation matrix
Returns
-----------
check : bool
True if matrix is a a transform with
only translation, scale, and rotation
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Optimized version of `compose_matrix` for just
scaling then translating.
Scalar args are broadcast to arrays of shape (3,)
Parameters
--------------
scale : float or (3,) float
Scale factor
translate : float or (3,) float
Translation
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Check to see if a matrix will invert triangles.
Parameters
-------------
matrix : (4, 4) float
Homogeneous transformation matrix
Returns
--------------
flip : bool
True if matrix will flip winding of triangles.
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