Rotation of 3D vector

Question

I have two vectors as Python lists and an angle  E g    v    3 5 0  axis    4 4 1  theta   1 2  radian   What is the best easiest way to get the resulting vector when rotating the v vector around the axis   The rotation should appear to be counter clockwise for an observer to whom the axis vector is pointing  This is called the right hand rule

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It can also be solved using quaternion theory   def angle axis quat theta  axis               Given an angle and an axis  it returns a quaternion              axis   np array axis    np linalg norm axis      return np append  np cos theta 2   np sin theta 2    axis   def mult quat q1  q2               Quaternion multiplication              q3   np copy q1      q3 0    q1 0  q2 0  - q1 1  q2 1  - q1 2  q2 2  - q1 3  q2 3      q3 1    q1 0  q2 1    q1 1  q2 0    q1 2  q2 3  - q1 3  q2 2      q3 2    q1 0  q2 2  - q1 1  q2 3    q1 2  q2 0    q1 3  q2 1      q3 3    q1 0  q2 3    q1 1  q2 2  - q1 2  q2 1    q1 3  q2 0      return q3  def rotate quat quat  vect               Rotate a vector with the rotation defined by a quaternion                Transfrom vect into an quaternion      vect   np append  0  vect        Normalize it     norm vect   np linalg norm vect      vect   vect norm vect       Computes the conjugate of quat     quat    np append quat 0  -quat 1          The result is given by  quat   vect   quat      res   mult quat quat  mult quat vect quat      norm vect     return res 1    v    3  5  0  axis    4  4  1  theta   1 2   print rotate quat angle axis quat theta  axis   v      2 74911638 4 77180932 1 91629719

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I made a fairly complete library of 3D mathematics for Python 2 3   It still does not use Cython  but relies heavily on the efficiency of numpy  You can find it here with pip   python 3  -m pip install math3d   Or have a look at my gitweb http   git automatics dyndns dk  p pymath3d git  and now also on github  https   github com mortlind pymath3d    Once installed  in python you may create the orientation object which can rotate vectors  or be part of transform objects  E g  the following code snippet composes an orientation that represents a rotation of 1 rad around the axis  1 2 3   applies it to the vector  4 5 6   and prints the result   import math3d as m3d r   m3d Orientation new axis angle  1 2 3   1  v   m3d Vector 4 5 6  print r   v    The output would be   lt Vector   2 53727  6 15234  5 71935  gt    This is more efficient  by a factor of approximately four  as far as I can time it  than the oneliner using scipy posted by B  M  above  However  it requires installation of my math3d package

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Use scipy s Rotation from rotvec    The argument is the rotation vector  a unit vector  multiplied by the rotation angle in rads   from scipy spatial transform import Rotation from numpy linalg import norm   v    3  5  0  axis    4  4  1  theta   1 2  axis   axis   norm axis     normalize the rotation vector first rot   Rotation from rotvec theta   axis   new v   rot apply v    print new v       results in  2 74911638 4 77180932 1 91629719    There are several more ways to use Rotation based on what data you have about the rotation    from quat Initialized from quaternions  from dcm Initialized from direction cosine matrices  from euler Initialized from Euler angles      Off-topic note  One line code is not necessarily better code as implied by some users

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Using the Euler-Rodrigues formula   import numpy as np import math  def rotation matrix axis  theta               Return the rotation matrix associated with counterclockwise rotation about     the given axis by theta radians              axis   np asarray axis      axis   axis   math sqrt np dot axis  axis       a   math cos theta   2 0      b  c  d   -axis   math sin theta   2 0      aa  bb  cc  dd   a   a  b   b  c   c  d   d     bc  ad  ac  ab  bd  cd   b   c  a   d  a   c  a   b  b   d  c   d     return np array   aa   bb - cc - dd  2    bc   ad   2    bd - ac                          2    bc - ad   aa   cc - bb - dd  2    cd   ab                          2    bd   ac   2    cd - ab   aa   dd - bb - cc     v    3  5  0  axis    4  4  1  theta   1 2   print np dot rotation matrix axis  theta   v        2 74911638  4 77180932  1 91629719

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I just wanted to mention that if speed is required  wrapping unutbu s code in scipy s weave inline and passing an already existing matrix as a parameter yields a 20-fold decrease in the running time   The code  in rotation matrix test py    import numpy as np import timeit  from math import cos  sin  sqrt import numpy random as nr  from scipy import weave  def rotation matrix weave axis  theta  mat   None       if mat    None          mat   np eye 3 3       support     include  lt math h gt       code               double x   sqrt axis 0    axis 0    axis 1    axis 1    axis 2    axis 2            double a   cos theta   2 0           double b   - axis 0    x    sin theta   2 0           double c   - axis 1    x    sin theta   2 0           double d   - axis 2    x    sin theta   2 0            mat 0    a a   b b - c c - d d          mat 1    2    b c - a d           mat 2    2    b d   a c            mat 3 1   0    2  b c a d           mat 3 1   1    a a c c-b b-d d          mat 3 1   2    2  c d-a b            mat 3 2   0    2  b d-a c           mat 3 2   1    2  c d a b           mat 3 2   2    a a d d-b b-c c               weave inline code    axis    theta    mat    support code   support  libraries     m         return mat  def rotation matrix numpy axis  theta       mat   np eye 3 3      axis   axis sqrt np dot axis  axis       a   cos theta 2       b  c  d   -axis sin theta 2        return np array   a a b b-c c-d d  2  b c-a d   2  b d a c                       2  b c a d   a a c c-b b-d d  2  c d-a b                       2  b d-a c   2  c d a b   a a d d-b b-c c      The timing    gt  gt  gt  import timeit  gt  gt  gt    gt  gt  gt  setup           import numpy as np     import numpy random as nr          from rotation matrix test import rotation matrix weave     from rotation matrix test import rotation matrix numpy          mat1   np eye 3 3      theta   nr random       axis   nr random 3           gt  gt  gt    gt  gt  gt  timeit repeat  rotation matrix weave axis  theta  mat1    setup setup  number 100000   0 36641597747802734  0 34883809089660645  0 3459300994873047   gt  gt  gt  timeit repeat  rotation matrix numpy axis  theta    setup setup  number 100000   7 180983066558838  7 172032117843628  7 180462837219238

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A one-liner  with numpy scipy functions   We use the following      let a be the unit vector along axis  i e  a   axis norm axis    and A   I    a be the skew-symmetric matrix associated to a  i e  the cross product of the identity matrix with a       then M   exp   A  is the rotation matrix    from numpy import cross  eye  dot from scipy linalg import expm  norm  def M axis  theta       return expm cross eye 3   axis norm axis  theta    v  axis  theta    3 5 0    4 4 1   1 2 M0   M axis  theta   print dot M0 v       2 74911638  4 77180932  1 91629719    expm  code here  computes the taylor series of the exponential   sum  k 0   20   frac 1  k      A  k   so it s time expensive  but readable and secure  It can be a good way if you have few rotations to do but a lot of vectors

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Take a look at http   vpython org contents docs visual VisualIntro html    It provides a vector class which has a method A rotate theta B   It also provides a helper function rotate A theta B  if you don t want to call the method on A   http   vpython org contents docs visual vector html

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I needed to rotate a 3D model around one of the three axes  x  y  z  in which that model was embedded and this was the top result for a search of how to do this in numpy  I used the following simple function   def rotate X  theta  axis  x         Rotate multidimensional array  X   theta  degrees around axis  axis       c  s   np cos theta   np sin theta    if axis     x   return np dot X  np array        1    0   0        0    c  -s        0    s   c          elif axis     y   return np dot X  np array        c   0   -s        0   1    0        s   0    c          elif axis     z   return np dot X  np array        c  -s   0         s   c   0         0   0   1

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Disclaimer  I am the author of this package  While special classes for rotations can be convenient  in some cases one needs rotation matrices  e g  for working with other libraries like the affine transform functions in scipy   To avoid everyone implementing their own little matrix generating functions  there exists a tiny pure python package which does nothing more than providing convenient rotation matrix generating functions  The package is on github  mgen  and can be installed via pip   pip install mgen   Example usage copied from the readme   import numpy as np np set printoptions suppress True   from mgen import rotation around axis from mgen import rotation from angles from mgen import rotation around x  matrix   rotation from angles  np pi 2  0  0    XYX   matrix dot  0  1  0     array  0   0   1     matrix   rotation around axis  1  0  0   np pi 2  matrix dot  0  1  0     array  0   0   1     matrix   rotation around x np pi 2  matrix dot  0  1  0     array  0   0   1      Note that the matrices are just regular numpy arrays  so no new data-structures are introduced when using this package

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Using pyquaternion is extremely simple  to install it  while still in python   run in your console   import pip  pip main   install   pyquaternion      Once installed     from pyquaternion import Quaternion   v    3 5 0    axis    4 4 1    theta   1 2  radian   rotated v   Quaternion axis axis angle theta  rotate v

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Here is an elegant method using quaternions that are blazingly fast  I can calculate 10 million rotations per second with appropriately vectorised numpy arrays  It relies on the quaternion extension to numpy found here   Quaternion Theory  A quaternion is a number with one real and 3 imaginary dimensions usually written as q   w   xi   yj   zk where  i    j    k  are imaginary dimensions  Just as a unit complex number  c  can represent all 2d rotations by c exp i   theta   a unit quaternion  q  can represent all 3d rotations by q exp p   where  p  is a pure imaginary quaternion set by your axis and angle   We start by converting your axis and angle to a quaternion whose imaginary dimensions are given by your axis of rotation  and whose magnitude is given by half the angle of rotation in radians  The 4 element vectors  w  x  y  z  are constructed as follows   import numpy as np import quaternion as quat  v    3 5 0  axis    4 4 1  theta   1 2  radian  vector   np array  0     v  rot axis   np array  0     axis  axis angle    theta 0 5    rot axis np linalg norm rot axis    First  a numpy array of 4 elements is constructed with the real component w 0  for both the vector to be rotated vector and the rotation axis rot axis  The axis angle representation is then constructed by normalizing then multiplying by half the desired angle theta  See here for why half the angle is required   Now create the quaternions v and qlog using the library  and get the unit rotation quaternion q by taking the exponential   vec   quat quaternion  v  qlog   quat quaternion  axis angle  q   np exp qlog    Finally  the rotation of the vector is calculated by the following operation   v prime   q   vec   np conjugate q   print v prime    quaternion 0 0  2 7491163  4 7718093  1 9162971    Now just discard the real element and you have your rotated vector    v prime vec   v prime imag     2 74911638 4 77180932 1 91629719  as a numpy array   Note that this method is particularly efficient if you have to rotate a vector through many sequential rotations  as the quaternion product can just be calculated as q   q1   q2   q3   q4         qn and then the vector is only rotated by  q  at the very end using v    q   v   conj q    This method gives you a seamless transformation between axis angle  lt ---  3d rotation operator simply by exp and log functions  yes log q  just returns the axis-angle representation    For further clarification of how quaternion multiplication etc  work  see here

[python] Rotation of 3D vector?

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