Rotation-Matrix

The main.py program loads a binary file containing 3D point cloud data formated as a list of 3-tuples (floats) corresponding to the (x,y,z) coordinates of a collection of points i.e. (point cloud). Then rotations about a user defined axis and magnitude (angle) are computed on each point. The original and resulting point clouds are visualized interactively.

Setup

Requires Python and the installation of following packages

• numpy
• matplotlib

Installation using pip

If using pip as your package manager you only need to type the following...

pip install -r requirements

To install these packages

If you continue to run into problems with matplotlib toolkits...

try the following

pip install -U matplotlib

If package import problems persist, please consider using a virtual environment.

Usage

C:\Users\rossm\devel\rotation-matrix> python .\main.py

Original Matrix

[[  50.    0.    0.    0.]
 [   1.    0. -209.  217.]
 [   2. -116. -250.  -81.]
 [   3.  -26.  228.  108.]
 [   4.  112.  214.  -45.]
 [   5.   95.   31.   77.]
 [   6.  -89.  241.  245.]
 [   7.   92.   77.  186.]
 ...
[  45.  100.  191.  -26.]
[  46.   16.  180. -143.]
[  47.   41. -243.   87.]
[  48.    7.   37.    3.]
[  49.   33.  195.  159.]
[  50.  -41.    8.  -29.]]

Rotated Matrix

[[  50.            0.            0.            0.        ]
 [   1.          194.79616897  -75.73277069  217.        ]
 [   2.          190.97627197 -198.70597259  -81.        ]
 [   3.         -221.92621322   58.38455179  108.        ]
 [   4.         -158.8722959   181.93293709  -45.        ]
 [   5.            5.53077501   99.77680356   77.        ]
 ...
[  46.         -161.9693114    80.13702118 -143.        ]
[  47.          241.34216582  -49.83933181   87.        ]
[  48.          -31.9489419    19.93151052    3.        ]
[  49.         -169.78981587  101.41705196  159.        ]
[  50.          -22.31298062  -35.31474049  -29.        ]]

Interactive Plots

Original and Rotated points are shown in different colors

You can click and drag to rotate the viewpoint

Information

See this pdf for explanation of the formula used to produce the rotation matrix function below

# set rotation axis to be the z-axis
axis = [0., 0., 1.]
# set rotation degree in radians
theta = 1.2

def rotation_matrix(axis, theta):
    """
    Return the rotation matrix associated with counterclockwise rotation about
    the user specified axis by theta radians.
    """
    # convert the input to an array
    axis = np.asarray(axis)
    # Get unit vector of our axis
    axis = axis/math.sqrt(np.dot(axis, axis))
    # take the cosine of out rotation degree in radians
    a = math.cos(theta/2.0)
    # get the rest rotation matrix components
    b, c, d = -axis*math.sin(theta/2.0)
    # create squared terms
    aa, bb, cc, dd = a*a, b*b, c*c, d*d
    # create cross terms
    bc, ad, ac, ab, bd, cd = b*c, a*d, a*c, a*b, b*d, c*d
    # return our rotation matrix

    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]])