13.2: Transformation Matrix

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Consider the following set of points:

%matplotlib inline
import matplotlib.pylab as plt

x = [0.0,  0.0,  2.0,  8.0, 10.0, 10.0, 8.0, 4.0, 3.0, 3.0, 4.0, 6.0, 7.0, 7.0, 10.0, 
     10.0,  8.0,  2.0, 0.0, 0.0, 2.0, 6.0, 7.0,  7.0,  6.0,  4.0,  3.0, 3.0, 0.0]
y = [0.0, -2.0, -4.0, -4.0, -2.0,  2.0, 4.0, 4.0, 5.0, 7.0, 8.0, 8.0, 7.0, 6.0,  6.0,
     8.0, 10.0, 10.0, 8.0, 4.0, 2.0, 2.0, 1.0, -1.0, -2.0, -2.0, -1.0, 0.0, 0.0]

plt.plot(x,y, color='green');
plt.axis('equal');

We can rotate these points around the origin by using the following simple set of equations:

\[ x \cos(\theta) - y \sin(\theta) = x_{rotated} \nonumber \]

\[ x \sin(\theta) + y \cos(\theta) = y_{rotated} \nonumber \]

This can be rewritten as the following system of matrices:

\[\begin{split}
\left[
\begin{matrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{matrix}
\right]
\left[
\begin{matrix}
x \\
y
\end{matrix}
\right]
=
\left[
\begin{matrix}
x_{rotated}\\
y_{rotated}
\end{matrix}
\right]
\end{split} \nonumber \]

We can rotate the points around the origin by \(\pi/4\) (i.e. \(45^o\)) as follows:

import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True) # Trick to make matrixes look nice in jupyter

points = np.matrix([x,y])

angle = np.pi/4
R = np.matrix([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]]);
sym.Matrix(R)

p=R*points

plt.plot(p[0].T,p[1].T);
plt.axis('equal');

#print(p[0].T)

We can even have a little fun and keep applying the same rotation over and over again.

# Apply R and plot 8 times
for i in range(0,8):
    p = R * p
    plt.plot(p[0].T,p[1].T);

plt.axis('equal');

Question

In the above code what does the T call in p[0].T do?