pycram.tf_transformations ========================= .. py:module:: pycram.tf_transformations .. autoapi-nested-parse:: This file was taken from> https://github.com/DLu/tf_transformations/tree/main Since it is not released for ROS1 we include it directly in PyCRAM This library is a reimplementation of the tf/transformations.py library. https://github.com/ros/geometry/blob/noetic-devel/tf/src/tf/transformations.py Original author: Christoph Gohlke Laboratory for Fluorescence Dynamics, University of California, Irvine Makes use of https://matthew-brett.github.io/transforms3d/ which is also a reimplementation of the Gohlke's work, but this maintains the API. Attributes ---------- .. autoapisummary:: pycram.tf_transformations.TRANSLATION_IDENTITY pycram.tf_transformations.ROTATION_IDENTITY pycram.tf_transformations.ZOOM_IDENTITY pycram.tf_transformations.SHEAR_IDENTITY pycram.tf_transformations._EPS pycram.tf_transformations._NEXT_AXIS pycram.tf_transformations._AXES2TUPLE pycram.tf_transformations._TUPLE2AXES Classes ------- .. autoapisummary:: pycram.tf_transformations.Arcball Functions --------- .. autoapisummary:: pycram.tf_transformations.identity_matrix pycram.tf_transformations.translation_matrix pycram.tf_transformations.translation_from_matrix pycram.tf_transformations.reflection_matrix pycram.tf_transformations.reflection_from_matrix pycram.tf_transformations.rotation_matrix pycram.tf_transformations.rotation_from_matrix pycram.tf_transformations.scale_matrix pycram.tf_transformations.scale_from_matrix pycram.tf_transformations.projection_matrix pycram.tf_transformations.projection_from_matrix pycram.tf_transformations.clip_matrix pycram.tf_transformations.shear_matrix pycram.tf_transformations.shear_from_matrix pycram.tf_transformations.decompose_matrix pycram.tf_transformations.compose_matrix pycram.tf_transformations.orthogonalization_matrix pycram.tf_transformations.superimposition_matrix pycram.tf_transformations.euler_matrix pycram.tf_transformations.euler_from_matrix pycram.tf_transformations.euler_from_quaternion pycram.tf_transformations._reorder_input_quaternion pycram.tf_transformations._reorder_output_quaternion pycram.tf_transformations.quaternion_from_euler pycram.tf_transformations.quaternion_about_axis pycram.tf_transformations.quaternion_matrix pycram.tf_transformations.quaternion_from_matrix pycram.tf_transformations.quaternion_multiply pycram.tf_transformations.quaternion_conjugate pycram.tf_transformations.quaternion_inverse pycram.tf_transformations.quaternion_slerp pycram.tf_transformations.random_quaternion pycram.tf_transformations.random_rotation_matrix pycram.tf_transformations.arcball_map_to_sphere pycram.tf_transformations.arcball_constrain_to_axis pycram.tf_transformations.arcball_nearest_axis pycram.tf_transformations.vector_norm pycram.tf_transformations.unit_vector pycram.tf_transformations.random_vector pycram.tf_transformations.inverse_matrix pycram.tf_transformations.concatenate_matrices pycram.tf_transformations.is_same_transform Module Contents --------------- .. py:data:: TRANSLATION_IDENTITY :value: [0.0, 0.0, 0.0] .. py:data:: ROTATION_IDENTITY .. py:data:: ZOOM_IDENTITY :value: [1.0, 1.0, 1.0] .. py:data:: SHEAR_IDENTITY :value: [0.0, 0.0, 0.0] .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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) .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: clip_matrix(left, right, bottom, top, near, far, perspective=False) 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 = numpy.random.rand(6) >>> frustum[1] += frustum[0] >>> frustum[3] += frustum[2] >>> frustum[5] += frustum[4] >>> M = clip_matrix(*frustum, perspective=False) >>> numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1.0]) array([-1., -1., -1., 1.]) >>> numpy.dot(M, [frustum[1], frustum[3], frustum[5], 1.0]) array([ 1., 1., 1., 1.]) >>> M = clip_matrix(*frustum, perspective=True) >>> v = numpy.dot(M, [frustum[0], frustum[2], frustum[4], 1.0]) >>> v / v[3] array([-1., -1., -1., 1.]) >>> v = numpy.dot(M, [frustum[1], frustum[3], frustum[4], 1.0]) >>> v / v[3] array([ 1., 1., -1., 1.]) .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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) .. py:function:: 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" .. py:function:: 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 .. py:function:: _reorder_input_quaternion(quaternion) Reorder quaternion to have w term first. .. py:function:: _reorder_output_quaternion(quaternion) Reorder quaternion to have w term last. .. py:function:: 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 .. py:function:: 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 .. py:function:: quaternion_matrix(quaternion) Return 4x4 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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,) .. py:function:: 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 .. py:class:: Arcball(initial=None) Bases: :py:obj:`object` 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() .. py:attribute:: _axis :value: None .. py:attribute:: _axes :value: None .. py:attribute:: _radius :value: 1.0 .. py:attribute:: _center :value: [0.0, 0.0] .. py:attribute:: _vdown .. py:attribute:: _constrain :value: False .. py:method:: 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. .. py:method:: setaxes(*axes) Set axes to constrain rotations. .. py:method:: setconstrain(constrain) Set state of constrain to axis mode. .. py:method:: getconstrain() Return state of constrain to axis mode. .. py:method:: down(point) Set initial cursor window coordinates and pick constrain-axis. .. py:method:: drag(point) Update current cursor window coordinates. .. py:method:: next(acceleration=0.0) Continue rotation in direction of last drag. .. py:method:: matrix() Return homogeneous rotation matrix. .. py:function:: arcball_map_to_sphere(point, center, radius) Return unit sphere coordinates from window coordinates. .. py:function:: arcball_constrain_to_axis(point, axis) Return sphere point perpendicular to axis. .. py:function:: arcball_nearest_axis(point, axes) Return axis, which arc is nearest to point. .. py:data:: _EPS .. py:data:: _NEXT_AXIS :value: [1, 2, 0, 1] .. py:data:: _AXES2TUPLE .. py:data:: _TUPLE2AXES .. py:function:: 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 .. py:function:: 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] .. py:function:: 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 .. py:function:: 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 .. py:function:: 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 .. py:function:: 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