[Numpy-discussion] Faster way to generate a rotation matrix?

[Robert Cimrman]

Jonathan Taylor wrote:
> Sorry.. obviously having some copy and paste trouble here.  The
> message should be as follows:
> 
> Hi,
> 
> I am doing optimization on a vector of rotation angles tx,ty and tz
> using scipy.optimize.fmin.  Unfortunately the function that I am
> optimizing needs the rotation matrix corresponding to this vector so
> it is getting constructed once for each iteration with new values.
>> >From profiling I can see that the function I am using to construct
> this rotation matrix is a bottleneck.  I am currently using:
> 
> def rotation(theta):
>    tx,ty,tz = theta
> 
>    Rx = np.array([[1,0,0], [0, cos(tx), -sin(tx)], [0, sin(tx), cos(tx)]])
>    Ry = np.array([[cos(ty), 0, -sin(ty)], [0, 1, 0], [sin(ty), 0, cos(ty)]])
>    Rz = np.array([[cos(tz), -sin(tz), 0], [sin(tz), cos(tz), 0], [0,0,1]])
> 
>    return np.dot(Rx, np.dot(Ry, Rz))
> 
> Is there a faster way to do this?  Perhaps I can do this faster with a
> small cython module, but this might be overkill?
> 
> Thanks for any help,
> Jonathan.

An alternative to specifying the rotation by the three angles tx,ty and 
tz could be creating directly the rotation matrix given an axis and an 
angle:

def make_axis_rotation_matrix(direction, angle):
     """
     Create a rotation matrix corresponding to the rotation around a general
     axis by a specified angle.

     R = dd^T + cos(a) (I - dd^T) + sin(a) skew(d)

     Parameters:

         angle : float a
         direction : array d
     """
     d = np.array(direction, dtype=np.float64)
     d /= np.linalg.norm(d)

     eye = np.eye(3, dtype=np.float64)
     ddt = np.outer(d, d)
     skew = np.array([[    0,  d[2],  -d[1]],
                      [-d[2],     0,  d[0]],
                      [d[1], -d[0],    0]], dtype=np.float64)

     mtx = ddt + np.cos(angle) * (eye - ddt) + np.sin(angle) * skew
     return mtx

r.