i dZddlZddlZddlZddlZddlZddlmZddl m Z ddl m Z dZdZd Zd Zd Zdhd Zd ZdidZdZ djdZdkdZdkdZdZdZdZ dldZdZdmdZdndZ dndZ!dndZ"dndZ#d Z$d!Z%dkd"Z&d#Z'd$Z(d%Z)d&Z*d'Z+dod(Z,dhd)Z-dhd*Z.Gd+d,e/Z0d-Z1d.Z2d/Z3ej4e5j6d0zZ7gd1Z8idd2d3d4d5d6d7d8d9d:d;d<d=d>d?d@dAdBdCdDdEdFdGdHdIdJdKdLdMdNdOdPdQdRdSdTdUdVdWdXdYdZZ9e:d[e9;DZd^Z?d_Z@d`ZAdaZBdpddZCdeZDeCdfdgZEdS)qa Homogeneous Transformation Matrices and Quaternions --- :mod:`MDAnalysis.lib.transformations` ============================================================================================== 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 :Version: 2010.05.10 :Licence: BSD 3-clause Requirements ------------ * `Python 2.6 or 3.1 `__ * `Numpy 1.4 `__ * `transformations.c 2010.04.10 `__ (optional implementation of some functions in C) 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 ``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". Use the transpose of transformation matrices for OpenGL ``glMultMatrixd()``. 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. .. rubric:: References .. footbibliography:: Examples -------- >>> from MDAnalysis.lib.transformations import * >>> import numpy as np >>> 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, 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 Functions --------- .. See `help(MDAnalysis.lib.transformations)` for a listing of functions or .. the online help. .. versionchanged:: 0.11.0 Transformations library moved from MDAnalysis.core.transformations to MDAnalysis.lib.transformations N)norm) no_copy_shim)anglecBtjdtjS)aAReturn 4x4 identity/unit matrix. >>> from MDAnalysis.lib.transformations import identity_matrix >>> import numpy as np >>> I = identity_matrix() >>> np.allclose(I, np.dot(I, I)) True >>> np.sum(I), np.trace(I) (4.0, 4.0) >>> np.allclose(I, np.identity(4, dtype=np.float64)) True dtype)npidentityfloat64h/srv/www/vhosts/g4struct/public_html/venv/lib/python3.11/site-packages/MDAnalysis/lib/transformations.pyidentity_matrixrs ;q + + ++rcPtjd}|dd|dddf<|S)zReturn matrix to translate by direction vector. >>> from MDAnalysis.lib.transformations import translation_matrix >>> import numpy as np >>> v = np.random.random(3) - 0.5 >>> np.allclose(v, translation_matrix(v)[:3, 3]) True rN)r r ) directionMs rtranslation_matrixrs/ AA!}Abqb!eH Hrcftj|ddddfS)aJReturn translation vector from translation matrix. >>> from MDAnalysis.lib.transformations import (translation_matrix, ... translation_from_matrix) >>> import numpy as np >>> v0 = np.random.random(3) - 0.5 >>> v1 = translation_from_matrix(translation_matrix(v0)) >>> np.allclose(v0, v1) True FcopyNr)r arrayrmatrixs rtranslation_from_matrixrs2 8F ' ' 'A . 3 3 5 55rct|dd}tjd}|ddddfxxdtj||zzcc<dtj|dd|z|z|dddf<|S)aReturn matrix to mirror at plane defined by point and normal vector. >>> from MDAnalysis.lib.transformations import reflection_matrix >>> import numpy as np >>> v0 = np.random.random(4) - 0.5 >>> v0[3] = 1.0 >>> 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 Nrr@) unit_vectorr r outerdot)pointnormalrs rreflection_matrixr%s* $ $F AAbqb"1"fIIIrx////IIIbfU2A2Y///69Abqb!eH HrcRtj|tjd}tj|ddddf\}}tjt tj|dzdkd}t|stdtj|dd|df }tj|\}}tjt tj|dz dkd}t|std tj|dd|d f }||dz}||fS) aReturn mirror plane point and normal vector from reflection matrix. >>> from MDAnalysis.lib.transformations import (reflection_matrix, ... reflection_from_matrix, is_same_transform) >>> import numpy as np >>> v0 = np.random.random(3) - 0.5 >>> v1 = np.random.random(3) - 0.5 >>> M0 = reflection_matrix(v0, v1) >>> point, normal = reflection_from_matrix(M0) >>> M1 = reflection_matrix(point, normal) >>> is_same_transform(M0, M1) True Fr rNr?:0yE>rz2no unit eigenvector corresponding to eigenvalue -11no unit eigenvector corresponding to eigenvalue 1) r rr linalgeigwhereabsreallen ValueErrorsqueeze)rrlVir$r#s rreflection_from_matrixr7s` rz666A 9==2A2rr6 # #DAq RWQZZ#%&&-..q1A q66OMNNN WQqqq!A$wZ ( ( * *F 9==  DAq RWQZZ#%&&-..q1A q66NLMMM GAaaa2hK ( ( * *E U1XE &=rctj|}tj|}t|dd}t j|ddfd|dfdd|fftj}|t j||d|z zz }||z}|t jd|d |df|dd|d f|d |ddfftjz }t jd }||ddddf<|Mt j|ddtjt }|t j ||z |dddf<|S) aReturn matrix to rotate about axis defined by point and direction. >>> from MDAnalysis.lib.transformations import (rotation_matrix, ... is_same_transform) >>> import random, math >>> import numpy as np >>> R = rotation_matrix(math.pi/2.0, [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*math.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*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 = np.identity(4, np.float64) >>> np.allclose(I, rotation_matrix(math.pi*2, direc)) True >>> np.allclose(2., np.trace(rotation_matrix(math.pi/2, ... direc, point))) True Nrr r(rrrr') mathsincosr r rr r!r rr")rrr#sinacosaRrs rrotation_matrixrAst: 8E??D 8E??DIbqbM**I  3  $  #t  j    A)Y ' '3: 66A I 9Q<-1 . q\31 .l]IaL# . j    A AAAbqb"1"fI rr"*<HHH26!U+++"1"a% Hrctj|tjd}|ddddf}tj|j\}}tjttj|dz dkd}t|stdtj|dd|d f }tj|\}}tjttj|dz dkd}t|stdtj|dd|d f }||dz}tj |dz d z } t|d dkr*|d | dz |dz|d zz|d z } nlt|d dkr*|d| dz |dz|d zz|d z } n)|d| dz |d z|d zz|dz } tj| | } | ||fS)a<Return rotation angle and axis from rotation matrix. >>> from MDAnalysis.lib.transformations import (rotation_matrix, ... is_same_transform, rotation_from_matrix) >>> import random, math >>> import numpy as np >>> angle = (random.random() - 0.5) * (2*math.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 Fr'Nrr(r)rr*r+rr:rrrrr:r:r)r rr r,r-Tr.r/r0r1r2r3tracer;atan2) rr@R33r4Wr6rQr#r?r>rs rrotation_from_matrixrLYsZ" rz666A BQBF)C 9==  DAq RWQZZ#%&&-..q1A q66NLMMM!!!QrU( $$,,..I 9==  DAq RWQZZ#%&&-..q1A q66NLMMM GAaaa2hK ( ( * *E U1XE HSMMC 3 &D 9Q<4 dGtczYq\1IaL@ @ aL Yq\  T ! ! dGtczYq\1IaL@ @ aL dGtczYq\1IaL@ @ aL JtT " "E )U ""rc|[tj|dddfd|ddfdd|dfdftj}|(|dd|dddf<|dddfxxd|z zcc<nt|dd}d|z }tjd}|ddddfxx|tj||zzcc<|*|tj|dd|z|z|dddf<|S)aVReturn matrix to scale by factor around origin in direction. Use factor -1 for point symmetry. >>> from MDAnalysis.lib.transformations import scale_matrix >>> import random >>> import numpy as np >>> v = (np.random.rand(4, 5) - 0.5) * 20.0 >>> v[3] = 1.0 >>> S = scale_matrix(-1.234) >>> np.allclose(np.dot(S, v)[:3], -1.234*v[:3]) True >>> factor = random.random() * 10 - 5 >>> origin = np.random.random(3) - 0.5 >>> direct = np.random.random(3) - 0.5 >>> S = scale_matrix(factor, origin) >>> S = scale_matrix(factor, origin, direct) Nr9r9r9r9r(r rr(r)r rr r r r!r")factororiginrrs r scale_matrixrQs1( Hc3'fc3'c63'$   *     bqbzAbqb!eH bqb!eHHHf $HHH "1" .. v KNN "1"bqb& Vbhy)<<<<  rr I!>!>>)KAbqb!eH Hrctj|tjd}|ddddf}tj|dz } tj|\}}tjttj||z dkdd}tj|dd|f }|t|z}n#t$r |dzdz }d}YnwxYwtj|\}}tjttj|d z dkd}t|std tj|dd|d f }||dz}|||fS) aReturn scaling factor, origin and direction from scaling matrix. >>> from MDAnalysis.lib.transformations import (scale_matrix, ... scale_from_matrix, is_same_transform) >>> import random >>> import numpy as np >>> factor = random.random() * 10 - 5 >>> origin = np.random.random(3) - 0.5 >>> direct = np.random.random(3) - 0.5 >>> S0 = scale_matrix(factor, origin) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True >>> S0 = scale_matrix(factor, origin, direct) >>> factor, origin, direction = scale_from_matrix(S0) >>> S1 = scale_matrix(factor, origin, direction) >>> is_same_transform(S0, S1) True Fr'Nrrr)r@r(,no eigenvector corresponding to eigenvalue 1r+)r rr rGr,r-r.r/r0r3 vector_norm IndexErrorr1r2) rrM33rOr4r5r6rrPs rscale_from_matrixrXs, rz666A BQBF)C Xc]]S F y}}S!!1 HSf,--4 5 5a 8 ;GAaaadG$$,,.. [+++ 3,#%  9==  DAq RWQZZ#%&&-..q1A q66IGHHH WQqqq!B%x[ ! ! ) ) + +F fQiF 69 $$sB*C33D  D Fctjd}tj|ddtjt}t |dd}|tj|ddtjd}tj||z |x|d<x|d<|d<|ddddfxxtj||zcc<|rL|ddddfxxtj||zcc<tj||||zz|dddf<ntj|||z|dddf<| |dddf<tj|||d <n|tj|ddtjt}tj||}|ddddfxxtj|||z zcc<|tj|||z z|dddf<nH|ddddfxxtj||zcc<tj|||z|dddf<|S) aReturn 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). >>> from MDAnalysis.lib.transformations import (projection_matrix, ... is_same_transform) >>> import numpy as np >>> 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) * 20.0 >>> v0[3] = 1.0 >>> v1 = np.dot(P, v0) >>> np.allclose(v1[1], v0[1]) True >>> np.allclose(v1[0], 3.0-v1[1]) True rNrr'Frrrrr:r:rr)r r rr rr r"r!)r#r$r perspectivepseudorscales rprojection_matrixraseH AA HU2A2Ybj| D D DE  $ $Fh{2A2bjuMMM &(f[5-@&&I&II$I!D'AdG "1"bqb& RXk6222  ; bqb"1"fIII&&11 1IIIveV,, f0DEAbqb!eHHveV,,{:Abqb!eH7!RaR%&f--$  H bqbM,   y&)) "1"bqb& RXi00588 uf 5 5 =>"1"a% "1"bqb& RXff--- 6%((61"1"a% Hrctj|tjd}|ddddf}tj|\}}tjt tj|dz dkd}|st|rtj|dd|df }||dz}tj|\}}tjt tj|dkd}t|std tj|dd|df }|t|z}tj|j \}}tjt tj|dkd}t|rOtj|dd|df } | t| z} || |ddfS||dddfStjt tj|dkd}t|std tj|dd|df }||dz}|dddf } |dddftj |dd| z } |r| | z} || d| |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. >>> from MDAnalysis.lib.transformations import (projection_matrix, ... projection_from_matrix, is_same_transform) >>> import numpy as np >>> 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 Fr'Nrr(r)rr+z,no eigenvector corresponding to eigenvalue 0z0no eigenvector not corresponding to eigenvalue 0)r rr r,r-r.r/r0r1r3r2rUrFr") rr_rrWr4r5r6r#rr$r^s rprojection_from_matrixrc(sF rz666A BQBF)C 9==  DAq RWQZZ#%&&-..q1A #8c!ff#8!!!QrU( $$,,.. qy}}S!!1 HS__t+ , ,Q /1vv MKLL LGAaaa1gJ''//11 [+++ y}}SU##1 HS__t+ , ,Q / q66 7WQqqq!A$wZ((0022F k&)) )F&)T58 8)T46 6 HS__t+ , ,Q /1vv B !!!QrU( $$,,.. qArrE(Ahbqb 6!:!::  " 6 !KfdK77rc||ks ||ks||krtd|re|tkrtdd|z}| ||z z d||z||z z dfd| ||z z ||z||z z dfdd||z ||z z ||z||z z fdf}n?d||z z dd||z||z z fdd||z z d||z||z z fddd||z z ||z||z z fdf}tj|tjS)aRReturn 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). >>> from MDAnalysis.lib.transformations import clip_matrix >>> import numpy as np >>> frustrum = np.random.rand(6) >>> frustrum[1] += frustrum[0] >>> frustrum[3] += frustrum[2] >>> frustrum[5] += frustrum[4] >>> M = clip_matrix(perspective=False, *frustrum) >>> np.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0]) array([-1., -1., -1., 1.]) >>> np.dot(M, [frustrum[1], frustrum[3], frustrum[5], 1.0]) array([1., 1., 1., 1.]) >>> M = clip_matrix(perspective=True, *frustrum) >>> v = np.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0]) >>> v / v[3] array([-1., -1., -1., 1.]) >>> v = np.dot(M, [frustrum[1], frustrum[3], frustrum[4], 1.0]) >>> v / v[3] array([ 1., 1., -1., 1.]) zinvalid frustrumzinvalid frustrum: near <= 0rr9)r9r9r9rNr )r2_EPSr rr ) leftrightbottomtopnearfarr^trs r clip_matrixrnus]H u}}# +,,, 4<<:;; ; $JR54< # 'F L 1"f %f v'F L #t}d 3QWd 5K L !  EDL !3edlte|-L M #v&cFlv|-L M #scDj)C$J4#:+F G   8ARZ ( ( ((rct|dd}t|dd}ttj||dkrt dt j|}tjd}|ddddfxx|tj||zz cc<| tj|dd|z|z|dddf<|S)amReturn 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. >>> from MDAnalysis.lib.transformations import shear_matrix >>> import random, math >>> import numpy as np >>> angle = (random.random() - 0.5) * 4*math.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(1.0, np.linalg.det(S)) True Nrgư>z/direction and normal vectors are not orthogonalr) r r/r r"r2r;tanr r!)rrr#r$rs r shear_matrixrqs. $ $FIbqbM**I 26&) $ $%%,,JKKK HUOOE AAbqb"1"fIII)V4444IIIvuRaRy&111I=Abqb!eH Hrctj|tjd}|ddddf}tj|\}}tjt tj|dz dkd}t|dkr"td |tj|dd|f j }d }d D]?\}}tj ||||} t| }||kr|}| } @| |z} tj|tjdz | } t| } | | z} t#j| } tj|\}}tjt tj|dz d kd}t|std tj|dd|df } | | dz} | | | | fS)aReturn shear angle, direction and plane from shear matrix. >>> from MDAnalysis.lib.transformations import (shear_matrix, ... shear_from_matrix, is_same_transform) >>> import random, math >>> import numpy as np >>> angle = (random.random() - 0.5) * 4*math.pi >>> direct = np.random.random(3) - 0.5 >>> point = np.random.random(3) - 0.5 >>> normal = np.cross(direct, np.random.random(3)) >>> S0 = shear_matrix(angle, direct, point, normal) >>> angle, direct, point, normal = shear_from_matrix(S0) >>> S1 = shear_matrix(angle, direct, point, normal) >>> is_same_transform(S0, S1) True Fr'Nrr(g-C6?rr:z2no two linear independent eigenvectors found {0!s}re)rrrDrr:r)rTr+)r rr r,r-r.r/r0r1r2formatr3rFcrossrUr"r r;atan)rrrWr4r5r6lenormi0i1nr$rrr#s rshear_from_matrixr|s$ rz666A BQBF)C 9==  DAq RWQZZ#%&&-..q1A 1vvzz @ G G J J    !!!Q$  ""$A F*B HQrUAbE " " NN v::FF fFsR[^^+V44I  " "E I Ie  E 9==  DAq RWQZZ#%&&-..q1A q66IGHHH GAaaa2hK ( ( * *E U1XE )UF **rc\tj|tjdj}t |dt krt d||dz}|}d|dddf<tj |st dtj d tj }gd }gd }tt |dddft krKtj |dddftj |j}d|dddf<n tjdtj }|dddf}d |dddf<|ddddf}t|d |d <|d xx|d zcc<tj |d |d |d <|d xx|d |d zzcc<t|d |d <|d xx|d zcc<|d xx|d zcc<tj |d |d|d <|dxx|d |d zzcc<tj |d |d|d<|dxx|d |dzzcc<t|d|d<|dxx|dzcc<|d dxx|dzcc<tj |d tj|d |dd kr |dz}|dz}t!j|d |d <t!j|d rIt!j|d|d|d <t!j|d|d|d<n*t!j|d |d|d <d|d<|||||fS)aReturn sequence of transformations from transformation matrix. matrix : array_like Non-degenerative homogeneous transformation matrix Return tuple of: scale : vector of 3 scaling factors shear : list of shear factors for x-y, x-z, y-z axes angles : list of Euler angles about static x, y, z axes translate : translation vector along x, y, z axes perspective : perspective partition of matrix Raise ValueError if matrix is of wrong type or degenerative. >>> from MDAnalysis.lib.transformations import (translation_matrix, ... decompose_matrix, scale_matrix, euler_matrix) >>> import numpy as np >>> T0 = translation_matrix((1, 2, 3)) >>> scale, shear, angles, trans, persp = decompose_matrix(T0) >>> T1 = translation_matrix(trans) >>> np.allclose(T0, T1) True >>> S = scale_matrix(0.123) >>> scale, shear, angles, trans, persp = decompose_matrix(S) >>> scale[0] 0.123 >>> R0 = euler_matrix(1, 2, 3) >>> scale, shear, angles, trans, persp = decompose_matrix(R0) >>> R1 = euler_matrix(*angles) >>> np.allclose(R0, R1) True Tr'r]zM[3, 3] is zerorrrrNrzmatrix is singular)rr )rrrrrr:r+rDrtr\rsrZrEr[r9)r rr rFr/rfr2rr,detzerosanyr"invrUrvr;asinr=rH) rrPr`shearanglesr^ translaterows rdecompose_matrixrsF rz5557A 1T7||d*+++4LA AAaaadG 9==  /-... HT , , ,E IIE YYF 3q!Qx==4   ?fQqqq!tWbimmAC&8&899 !!!Q$h|2:>>> !RaR% IAa!eH BQBF)..  C3q6""E!HFFFeAhFFFvc!fc!f%%E!HFFFc!fuQxFFF3q6""E!HFFFeAhFFF !HHHaHHHvc!fc!f%%E!HFFFc!fuQxFFFvc!fc!f%%E!HFFFc!fuQxFFF3q6""E!HFFFeAhFFF !""IIIqIII vc!fbhs1vs1v..//!33   r  3t9*%%F1I xq Js4y#d)44q Js4y#d)44q JD z3t955q q %K 77rctjd}|:tjd}|dd|dddf<tj||}|:tjd}|dd|dddf<tj||}|9t|d|d|dd}tj||}|Jtjd} |d| d<|d| d <|d| d <tj|| }|Jtjd} |d| d <|d| d <|d| d <tj|| }||dz}|S)aReturn 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 >>> from MDAnalysis.lib.transformations import (compose_matrix, ... decompose_matrix, is_same_transform) >>> import math >>> import numpy as np >>> scale = np.random.random(3) - 0.5 >>> shear = np.random.random(3) - 0.5 >>> angles = (np.random.random(3) - 0.5) * (2*math.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 rNrrrr:sxyzrtrDrsrZr[r\r])r r r" euler_matrix) r`rrrr^rrrFr@ZSs rcompose_matrixr]s_< AA KNNbqb/!QQQ$ F1aLL KNNRaR="1"a% F1aLL  F1Ivay& A A F1aLL  KNN($($($ F1aLL  KNN($($($ F1aLL4LA Hrcn|\}}}tj|}tj|\}}}tj|\}} } || z| z ||zz } tj||zt jd| | zz zdddf| |z| z||zddf|| z||z|dfdftjS)aReturn orthogonalization matrix for crystallographic cell coordinates. Angles are expected in degrees. The de-orthogonalization matrix is the inverse. >>> from MDAnalysis.lib.transformations import orthogonalization_matrix >>> import numpy as np >>> 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 r(r9rNr )r radiansr<r=rr;sqrtr ) lengthsrabcr>sinb_r?cosbcosgcos rorthogonalization_matrixrs"GAq! Z  FF6NNMD$vf~~D$ +  -B 8 X #R-00 0#sC @R$Y^QXsC 0 Xq4xC (  j   rTc tj|tjtdd}tj|tjtdd}|j|jks|jddkrt dtj|d}tj|d}||ddz }||ddz }|rtj tj ||j \}}}tj ||} tj | dkr?| tj |dddf|dddfd zz} |d xxd zcc<tjd } | | ddddf<ntjd ||\} } } tjd |tj|d d\}}}tjd |tj|dd\}}}| | z| zdddf||z | | z | z ddf||z ||z| | z| z df||z ||z||z| | z | zff}tj|\}}|ddtj|f}|t'|z}t)|} |rS| ddddfxxt+jtjd||tjd||z zcc<|| dddf<tjd }| |dddf<tj | |} | S)aReturn 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). >>> from MDAnalysis.lib.transformations import (superimposition_matrix, ... random_rotation_matrix, scale_matrix, translation_matrix, ... concatenate_matrices) >>> import random >>> import numpy as np >>> 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) * 20.0 >>> v0[3] = 1.0 >>> 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(0.0, 1e-9, 300).reshape(3, -1) >>> M = superimposition_matrix(v0, v1, scaling=True) >>> np.allclose(v1, np.dot(M, v0)) True >>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False) >>> np.allclose(v1, np.dot(M, v0)) True >>> v = np.empty((4, 100, 3), dtype=np.float64) >>> v[:, :, 0] = v0 >>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False) >>> np.allclose(v1, np.dot(M, v[:, :, 0])) True r'Nrrz&vector sets are of wrong shape or typeaxisr9r:rr+rerzij,ij->irzij,ij->)r rr rshaper2meanreshaper,svdr"rFrr!r einsumrolleighargmaxrUquaternion_matrixr;r)v0v1scalingusesvdt0t1usvhr@rxxyyzzxyyzzxxzyxzyNr4r5qrFs rsuperimposition_matrixrsfh "BJ\ : : :2A2 >B "BJ\ : : :2A2 >B x28rx{QABBB !   B !   B bjjA B bjjA B !9==BD!1!1221b F1bMM 9==  c ! ! !AAAqD'2ad8c>22 2A bEEETMEEE KNN"1"bqb& Yz2r22 BYz2rwr2A/F/F/FGG BYz2rwr2A/F/F/FGG B "Wr\3S ) "Wb2glC - "Wb2gsRx"}c 2 "Wb2grBwb2 6  y~~a  1 aaa1o  [^^ a   "1"bqb& TY IiR ( (29YB+G+G G   Abqb!eH AAsAbqb!eH q! A Hrrc t|\}}}}n+#ttf$rt|}|\}}}}YnwxYw|} t| |z} t| |z dz} |r||}}|r | | | }}}t j|t j|t j|}} } t j|t j|t j|}}}||z||z}}| |z| |z}}tj d}|rg||| | f<| | z|| | f<| |z|| | f<| |z|| | f<| |z|z|| | f<| |z|z || | f<| |z|| | f<||z|z|| | f<||z|z || | f<nd||z|| | f<| |z|z || | f<| |z|z|| | f<||z|| | f<| |z|z|| | f<| |z|z || | f<| || | f<|| z|| | f<||z|| | f<|S)a 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 >>> from MDAnalysis.lib.transformations import (euler_matrix, ... _AXES2TUPLE, _TUPLE2AXES) >>> import math >>> import numpy as np >>> 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 = (4.0*math.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) rr) _AXES2TUPLEAttributeErrorKeyError _TUPLE2AXES _NEXT_AXISr;r<r=r r )aiajakaxes firstaxisparity repetitionframerr6jksisjskcicjckcccsscssrs rrr$s.4/:4/@, 6:uu H %444  /3, 6:uuu4 A1v:A1v:>"A RB #S2#sB"tx||TXb\\BB"tx||TXb\\BB "Wb2gB "Wb2gB AA!Q$r'!Q$r'!Q$r'!Q$#(R-!Q$#(R-!Q$#(!Q$r'B,!Q$r'B,!Q$r'!Q$r'B,!Q$r'B,!Q$r'!Q$r'B,!Q$r'B,!Q$#!Q$r'!Q$r'!Q$ Hs %==cV t|\}}}}n+#ttf$rt|}|\}}}}YnwxYw|}t ||z}t ||z dz} t j|t jdddddf} |rtj | ||f| ||fz| || f| || fzz} | tkrjtj | ||f| || f} tj | | ||f} tj | ||f| | |f }n=tj | || f | ||f} tj | | ||f} d}ntj | ||f| ||fz| ||f| ||fzz}|tkritj | | |f| | | f} tj | | |f |} tj | ||f| ||f}nFtj | || f | ||f} tj | | |f |} d}|r | | | }} } |r|| }} | | |fS)a1Return Euler angles from rotation matrix for specified axis sequence. axes : One of 24 axis sequences as string or encoded tuple Note that many Euler angle triplets can describe one matrix. >>> from MDAnalysis.lib.transformations import (euler_matrix, ... euler_from_matrix, _AXES2TUPLE) >>> import math >>> import numpy as np >>> R0 = euler_matrix(1, 2, 3, 'syxz') >>> al, be, ga = euler_from_matrix(R0, 'syxz') >>> R1 = euler_matrix(al, be, ga, 'syxz') >>> np.allclose(R0, R1) True >>> angles = (4.0*math.pi) * (np.random.random(3) - 0.5) >>> for axes in _AXES2TUPLE.keys(): ... R0 = euler_matrix(axes=axes, *angles) ... R1 = euler_matrix(axes=axes, *euler_from_matrix(R0, axes)) ... if not np.allclose(R0, R1): print(axes, "failed") rFr'Nrr9) rlowerrrrrr rr r;rrfrH)rrrrrrrr6rrrsyaxayazcys reuler_from_matrixrgs.4/:4::<"A rz666rr2A2v>A YqAw1a4(1QT7Qq!tW+<< = = 99AadGQq!tW--BB!Q$((BAadGa1gX..BBQq!tWHa1g..BB!Q$((BBB YqAw1a4(1QT7Qq!tW+<< = = 99AadGQq!tW--BQq!tWHb))BAadGQq!tW--BBQq!tWHa1g..BQq!tWHb))BB #S2#sB RB r2:$'%AAc<tt||S)a&Return Euler angles from quaternion for specified axis sequence. >>> from MDAnalysis.lib.transformations import euler_from_quaternion >>> import numpy as np >>> angles = euler_from_quaternion([0.99810947, 0.06146124, 0, 0]) >>> np.allclose(angles, [0.123, 0, 0]) True )rr) quaternionrs reuler_from_quaternionrs .z::D A AArc t|\}}}}n+#ttf$rt|}|\}}}}YnwxYw|dz} t | |zdz dz} t | |z dz} |r||}}|r| }|dz}|dz}|dz}t j|} t j|} t j|}t j|}t j|}t j|}| |z}| |z}| |z}| |z}tj dtj }|r-|||z z|d<|||zz|| <|||zz|| <|||z z|| <n8||z||zz|d<||z||zz || <||z||zz|| <||z||zz || <|r|| xxdzcc<|S)axReturn quaternion from Euler angles and axis sequence. ai, aj, ak : Euler's roll, pitch and yaw angles axes : One of 24 axis sequences as string or encoded tuple >>> from MDAnalysis.lib.transformations import quaternion_from_euler >>> q = quaternion_from_euler(1, 2, 3, 'ryxz') >>> np.allclose(q, [0.435953, 0.310622, -0.718287, 0.444435]) True rrrr rr+) rrrrrrr;r=r<r emptyr )rrrrrrrrrr6rrrrrrrrrrrrrs rquaternion_from_eulerrs"4/:4::<"Q&A1v:"A RB S#IB#IB#IB "B "B "B "B "B "B bB bB bB bB$bj111J *b2g 1 b2g 1 b2g 1 b2g 1 R"r') 1 R"r') 1 R"r') 1 R"r') 1  1  rc*tjdtj}|d|d<|d|d<|d|d<t|}|tkr|t j|dz |z z}t j|dz |d<|S)a Return quaternion for rotation about axis. >>> from MDAnalysis.lib.transformations import quaternion_about_axis >>> import numpy as np >>> q = quaternion_about_axis(0.123, (1, 0, 0)) >>> np.allclose(q, [0.99810947, 0.06146124, 0, 0]) True rr rrr:rr)r rr rUrfr;r<r=)rrrqlens rquaternion_about_axisrs$bj111JGJqMGJqMGJqM z " "D d{{dhus{++d22 HUS[))JqM rc tj|ddtjd}tj||}|tkrtjdS|t jd|z z}tj||}tjd|dz |dz |d |d z |d |d zd f|d |d zd|dz |dz |d|dz d f|d |d z |d|dzd|dz |dz d fdftjS)aReturn homogeneous rotation matrix from quaternion. >>> from MDAnalysis.lib.transformations import (identity_matrix, ... quaternion_matrix, rotation_matrix) >>> import numpy as np >>> 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, identity_matrix()) True >>> M = quaternion_matrix([0, 1, 0, 0]) >>> np.allclose(M, np.diag([1, -1, -1, 1])) True NrTr'rr(r\r]rt)rr)rrr:rr9r[)r:rrCrNr ) r rr r"rfr r;rr!)rrnqs rrrsU" BQBrz===A 1B Dyy{1~~38  A AA 8ag $'$!D'!$!D'!  $!D'!ag $'$!D'!  $!D'!$!D'!ag $'   !' *j-   rc tj|tjtddddf}|rTtjdtj}tj|}||dkrB||d<|d|d z |d <|d |d z |d <|d|dz |d<nd\}}}|d|dkrd\}}}|d|||fkrd\}}}|||f|||f|||fzz |dz}|||<|||f|||fz||<|||f|||fz||<|||f|||fz |d <|dt j||dzz z}n|d}|d } |d } |d} |d} |d} |d }|d}|d}tj|| z |z dddf| | z| |z |z ddf| |z| |z||z | z df|| z | |z | | z || z|zff}|dz}tj |\}}|gdtj |f}|ddkr|dz}|S)aReturn quaternion from rotation matrix. If isprecise=True, the input matrix is assumed to be a precise rotation matrix and a faster algorithm is used. >>> from MDAnalysis.lib.transformations import (identity_matrix, ... quaternion_from_matrix, rotation_matrix, random_rotation_matrix, ... is_same_transform, quaternion_matrix) >>> import numpy as np >>> q = quaternion_from_matrix(identity_matrix(), 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 r'Nrrr r]rrCrsrrDrr:rErtr)rr:rr[rZ)r:rrr\)rrr:g?r9rS)rrrr:re) r rr rrrGr;rr,rr)r ispreciserrrmr6rrm00m01m02m10m11m12m20m21m22Kr4r5s rquaternion_from_matrixr2sH rz ===bqb"1"fEA** HT , , , HQKK qw;;AaDT7QtW$AaDT7QtW$AaDT7QtW$AaDDGAq!w4  !1aw1a4  !1a!Q$1QT7Qq!tW,-$7AAaDQT7Qq!tW$AaDQT7Qq!tW$AaDQT7Qq!tW$AaD S49Q4[)) ))ggggggggg HsS#sC0sC#IOS#6sC#IsSy3<sC#IsSy#)c/B     Sy~~a  1 lllBIaLL( )tczz T  Hrc|\}}}}|\}}}} tj| |z||zz | |zz ||zz||z||zz| |zz ||zz| |z||zz| |zz||zz||z||zz | |zz||zzftjS)aReturn multiplication of two quaternions. >>> from MDAnalysis.lib.transformations import quaternion_multiply >>> import numpy as np >>> q = quaternion_multiply([4, 1, -2, 3], [8, -5, 6, 7]) >>> np.allclose(q, [28, -44, -14, 48]) True r r rr ) quaternion1 quaternion0w0x0y0z0w1x1y1z1s rquaternion_multiplyrs!NBB NBB 8 C"HrBw b (27 2 Gb2g R '"r' 1 C"HrBw b (27 2 Gb2g R '"r' 1  j   rctj|d|d |d |d ftjS)a,Return conjugate of quaternion. >>> from MDAnalysis.lib.transformations import (random_quaternion, ... quaternion_conjugate) >>> import numpy as np >>> q0 = random_quaternion() >>> q1 = quaternion_conjugate(q0) >>> q1[0] == q0[0] and all(q1[1:] == -q0[1:]) True rrr:rr rrs rquaternion_conjugater sD 8 AAAAGj   rcLt|tj||z S)a3Return inverse of quaternion. >>> from MDAnalysis.lib.transformations import (random_quaternion, ... quaternion_inverse) >>> import numpy as np >>> q0 = random_quaternion() >>> q1 = quaternion_inverse(q0) >>> np.allclose(quaternion_multiply(q0, q1), [1, 0, 0, 0]) True )r r r"r s rquaternion_inverser s# + +bfZ.L.L LLrc|dS)zReturn real part of quaternion. >>> from MDAnalysis.lib.transformations import quaternion_real >>> quaternion_real([3.0, 0.0, 1.0, 2.0]) 3.0 rrr s rquaternion_realrs a=rc|ddS)zReturn imaginary part of quaternion. >>> from MDAnalysis.lib.transformations import quaternion_imag >>> quaternion_imag([3.0, 0.0, 1.0, 2.0]) [0.0, 1.0, 2.0] rrrr s rquaternion_imagrs ac?rcNt|dd}t|dd}|dkr|S|dkr|Stj||}tt|dz tkr|S|r|dkr| }|dz}t j||t jzz}t|tkr|Sdt j|z } |t jd|z |z| zz}|t j||z| zz}||z }|S)aReturn spherical linear interpolation between two quaternions. >>> from MDAnalysis.lib.transformations import (random_quaternion, ... quaternion_slerp) >>> import math >>> import numpy as np >>> q0 = random_quaternion() >>> q1 = random_quaternion() >>> q = quaternion_slerp(q0, q1, 0.0) >>> np.allclose(q, q0) True >>> q = quaternion_slerp(q0, q1, 1.0, 1) >>> np.allclose(q, q1) True >>> q = quaternion_slerp(q0, q1, 0.5) >>> angle = math.acos(np.dot(q0, q)) >>> np.allclose(2.0, math.acos(np.dot(q0, q1)) / angle) or \ ... np.allclose(2.0, math.acos(-np.dot(q0, q1)) / angle) True Nrr9r(re) r r r"r/rfr;acospir<) quat0quat1fractionspin shortestpathq0q1drisins rquaternion_slerprs), U2A2Y  B U2A2Y  B3 S  r2A 3q66C<4 C B d  IaLL4$'> )E 5zzD % D$(C(Ne+ , ,t 33B$(8e# $ $t ++B"HB Irc| tjd}nt|dksJtjd|dz }tj|d}t jdz}||dz}||dz}tjtj||ztj ||ztj||ztj ||zftj S) aReturn uniform random unit quaternion. rand: array like or None Three independent random variables that are uniformly distributed between 0 and 1. >>> from MDAnalysis.lib.transformations import (random_quaternion, ... vector_norm) >>> import numpy as np >>> q = random_quaternion() >>> np.allclose(1.0, vector_norm(q)) True >>> q = random_quaternion(np.random.random(3)) >>> len(q.shape), q.shape[0]==4 (1, True) Nrr(rrrr:r ) r randomrandr1rr;rrr=r<r )r!r1r2pi2rt2s rrandom_quaternionr&s$ |y~~a  4yyA~~~~ tAw  B a  B 'C-C tAwB tAwB 8 F2JJO F2JJO F2JJO F2JJO  j   rc:tt|S)aReturn uniform random rotation matrix. rnd: array like Three independent random variables that are uniformly distributed between 0 and 1 for each returned quaternion. >>> from MDAnalysis.lib.transformations import random_rotation_matrix >>> import numpy as np >>> R = random_rotation_matrix() >>> np.allclose(np.dot(R.T, R), np.identity(4)) True )rr&)r!s rrandom_rotation_matrixr()s .t44 5 55rcLeZdZdZd dZdZdZdZdZdZ d Z dd Z d Z dS)ArcballaVirtual Trackball Control. >>> from MDAnalysis.lib.transformations import Arcball >>> import numpy as np >>> 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.setconstrain(True) >>> ball.down([400, 200]) >>> ball.drag([200, 400]) >>> R = ball.matrix() >>> np.allclose(np.sum(R), 0.2055924) True >>> ball.next() Ncd|_d|_d|_ddg|_t jgdtj|_d|_|(t jgdtj|_ ntt j|tj}|j dkrt||_ n4|j d kr|t|z}||_ ntd |j x|_|_dS) z`Initialize virtual trackball control. initial : quaternion or rotation matrix Nr(r9)rrrr Frrrr)rrrz"initial not a quaternion or matrix)_axis_axes_radius_centerr rr _vdown _constrain_qdownrrrUr2_qnow_qpre)selfinitials r__init__zArcball.__init__Ts    Sz hyyy ;;;  ?(<<| d|_dSd|D|_dS)z Set axes to constrain rotations.Nc,g|]}t|Sr)r ).0rs r z#Arcball.setaxes..s ===+d++===r)r.)r6rs rsetaxeszArcball.setaxes|s+ <DJJJ=====DJJJrc|du|_dS)z$Set state of constrain to axis mode.TNr2)r6 constrains r setconstrainzArcball.setconstrains#t+rc|jS)z'Return state of constrain to axis mode.rDr6s r getconstrainzArcball.getconstrains rct||j|j|_|jx|_|_|jrG|j@t|j|j|_ t|j|j |_dSd|_ dS)z>Set initial cursor window coordinates and pick constrain-axis.N) arcball_map_to_spherer0r/r1r4r3r5r2r.arcball_nearest_axisr-arcball_constrain_to_axis)r6r#s rdownz Arcball.downss+E4<NN #':- dj ? tz5-dk4:FFDJ3DKLLDKKKDJJJrct||j|j}|jt ||j}|j|_tj|j |}tj ||tkr|j |_dStj |j ||d|d|dg}t||j |_dS)z)Update current cursor window coordinates.Nrrr:)rKr0r/r-rMr4r5r rvr1r"rfr3r)r6r#vnowrmrs rdragz Arcball.drags$UDL$,GG : !,T4:>>DZ HT[$ ' ' 6!Q<<$  DJJJ T**AaD!A$!=A,Q <>>,,,   === //// -----rr*cJtj|d|dz |z |d|dz |z dftj}|d|dz|d|dzz}|dkr|tj|z}ntjd|z |d<|S)z7Return unit sphere coordinates from window coordinates.rrr9r r(r:)r rr r;r)r#r;r<vr{s rrKrKs  1Xq !V + AYq !V +  j    A !qt adQqTk!A3ww TYq\\yq!!! Hrctj|tjd}tj|tjd}||tj||zz}t |}|t kr|ddkr|dz}||z}|S|ddkr"tjgdtjSt |d  |d d gS) z*Return sphere point perpendicular to axis.Tr'r:r9rer(rrrr rr)r rr r"rUrfr )r#rr\rr{s rrMrMs bjt444A RZd333ARVAq\\ AAA4xx Q4#:: IA Qts{{x 4444 1qtQ' ( ((rctj|tjd}d}d}|D]/}tjt |||}||kr|}|}0|S)z+Return axis, which arc is nearest to point.Fr'Nre)r rr r"rM)r#rnearestmxrrms rrLrLsh HU"*5 9 9 9EG B F,UD995 A A r66GB Nrg@)rr:rr)rrrrsxyx)rrrrsxzy)rrrrsxzx)rrrrsyzxr,syzy)rrrrsyxz)rrrrsyxy)rrrrszxy)r:rrrszxz)r:rrrszyx)r:rrrszyz)r:rrrrzyxr~rxyx)rrrrryzx)rrrrrxzx)rrrrrxzy)rrrr)rrrr)rrrr)rrrr)r:rrr)r:rrr)r:rrr)r:rrr)ryzyrzxyryxyryxzrzxzrxyzrzyzc#$K|] \}}||fV dSrVr)r@rr\s r rzs*::daAq6::::::rctj|tjd}|v|jdkr't jtj||S||z}tjtj||}tj|||S||z}tj|||tj||dS)aReturn length, i.e. eucledian norm, of ndarray along axis. >>> from MDAnalysis.lib.transformations import vector_norm >>> import numpy as np >>> 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.empty((5, 3), dtype=np.float64) >>> vector_norm(v, axis=1, out=n) >>> np.allclose(n, np.sqrt(np.sum(v*v, axis=1))) True >>> vector_norm([]) 0.0 >>> vector_norm([1.0]) 1.0 Tr'Nrr)rout) r rr ndimr;rr" atleast_1dsum)datarr|s rrUrUs6 8D  6 6 6D { 9>>9RVD$//00 0  mBF4d33344 S    t$C(((( Src|Ytj|tjd}|jdkr,|t jtj||z}|Sn!||urtj|d|dd<|}tjtj||z|}tj|||tj ||}||z}||SdS)a0Return ndarray normalized by length, i.e. eucledian norm, along axis. >>> from MDAnalysis.lib.transformations import unit_vector >>> import numpy as np >>> 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.empty((5, 4, 3), dtype=np.float64) >>> unit_vector(v0, axis=1, out=v1) >>> np.allclose(v1, v2) True >>> list(unit_vector([])) [] >>> list(unit_vector([1.0])) [1.0] NTr'rFr) r rr r}r;rr"r~r expand_dims)rrr|lengths rr r s8 {xBJT::: 9>> DIbfT40011 1DK  d??Xd///CF ]26$+t44 5 5FGFF --FND { {rc@tj|S)a_Return array of random doubles in the half-open interval [0.0, 1.0). >>> from MDAnalysis.lib.transformations import random_vector >>> import numpy as np >>> v = random_vector(10000) >>> np.all(v >= 0.0) and np.all(v < 1.0) True >>> v0 = random_vector(10) >>> v1 = random_vector(10) >>> np.any(v0 == v1) False )r r )sizes r random_vectorrMs 9  D ! !!rc@tj|S)aReturn inverse of square transformation matrix. >>> from MDAnalysis.lib.transformations import (random_rotation_matrix, ... inverse_matrix) >>> import numpy as np >>> 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,rrs rinverse_matrixr^s 9==  rcbtjd}|D]}tj||}|S)a]Return concatenation of series of transformation matrices. >>> from MDAnalysis.lib.transformations import concatenate_matrices >>> import numpy as np >>> 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")matricesrr6s rconcatenate_matricesrqs6 AA  F1aLL Hrctj|tjd}||dz}tj|tjd}||dz}tj||S)aVReturn True if two matrices perform same transformation. >>> from MDAnalysis.lib.transformations import (is_same_transform, ... random_rotation_matrix) >>> import numpy as np >>> is_same_transform(np.identity(4), np.identity(4)) True >>> is_same_transform(np.identity(4), random_rotation_matrix()) False Tr'r])r rr allclose)matrix0matrix1s ris_same_transformrsbhwbjt<<tj|rtj d|zYdSYdSwxYw)zTry import all public attributes from module into global namespace. Existing attributes with name clashes are renamed with prefix. Attributes starting with underscore are ignored by default. Return True on successful import. zno Python implementation of Tzfailed to import module N)syspathappendosdirname__file__ __import__popdir startswithglobalswarningswarngetattr ImportError) module_namerprefixignoremoduleattrs r_import_modulersXHOOBGOOH--...K((  KK 4 4D $//&11  I799$$/6yyGIIftm,,IM"@4"GHHH%fd33GIIdOOt DDD    D M4{B C C C C C C D D DDsD AEEctj||rtjgdStj||}|tj|z S)aReturn the rotation axis to rotate vector a into b. Parameters ---------- a, b : array_like two vectors Returns ------- c : np.ndarray vector to rotate a into b Note ---- If a == b this will always return [1, 0, 0] r^)r rrrvr,r)rrrs rrotaxisrsT& {1a#x """ AA ry~~a   r_transformationszrestructuredtext enrV)NN)NNF)F)NNNNN)FT)r)rT)Trr)FrZr;rrrnumpyr numpy.linalgrMDAnalysis.lib.utilrmdamathrvecanglerrrr%r7rArLrQrXrarcrnrqr|rrrrrrrrrrrrr r rrrr&r(objectr*rKrMrLfinfor:epsrfrrdictitemsrrUr rrrrrr __docformat__rrrrs7DAAF ,,,,,,&&&&&&,,,"     6 6 6   8@9 9 9 9 x/#/#/#d* * * * Z*%*%*%\=B@ @ @ @ FJ8J8J8J8Z7)7)7)7)t   D0+0+0+fV8V8V8tFJ7 7 7 7 tBi i i i X@ @ @ @ F;;;;| B B B B7777t*...bS S S S l.$ M M M****Z####L6666"s-s-s-s-s-fs-s-s-l   $ ) ) )   rxS \\   L  , 06   L  , 06   L !, 17   L !, 17   L  !,  17   L  !,  9E , ,     d::k&7&7&9&9::::: &&&&R++++\""""!!!&   $)))&B!!!2!"""& r