Linear Transformations

In this exercise, you will apply matrices to transform vectors.

Clock

Let’s define a vector for a clock hand with an x and y coordinate:

\[\begin{split}\vec{a} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}\end{split}\]

and

import numpy as np
from matplotlib import pyplot as plt

a = np.array([0, 2])

The following code plots the clock:

fig, ax = plt.subplots(figsize=(4, 4))
ax.add_patch( plt.Circle((0, 0), 4.2, color='#cccccc'))
ax.plot([3, 4], [0, 0], linewidth=3, color="black")
ax.plot([-3, -4], [0, 0], linewidth=3, color="black")
ax.plot([0, 0], [3, 4], linewidth=3, color="black")
ax.plot([0, 0], [-3, -4], linewidth=3, color="black")

# draw the clock
ax.plot([0, a[0]], [0, a[1]], linewidth=5)

plt.axis([-5, 5, -5, 5])

Rotation

For linear transformations, use matrices. As long as the number of dimensions is the same before and after the transformation, it is a square matrix.

The most simple linear transformation is a rotation by 90°:

ROT = np.array([
    [0, 1],
    [-1, 0]]
)

To apply the rotation, use a matrix-vector product:

a = np.dot(ROT, a)

Then plot the clock again to see the change.

You can execute the dot product multiple times for multiple rotations.

Rotation by other angles

You can rotate by arbitrary angles using trigonometric functions:

from math import sin, cos, radians

angle = radians(-45)

ROT = np.array([
    [cos(angle), sin(angle)],
    [-sin(angle), cos(angle)]
])

Try the following:

• rotate the hand by half an hour

• move the clocks hand back 10 minutes

Stretch

Another type of linear transformations is a stretching operation. It scales up the coordinates proportionally:

STRETCH = np.array([
    [2, 0],
    [0, 2]
    ])
a = np.dot(STRETCH, a)

Now shrink the hand again.

Shear

The last operation modifies one coordinate but keeps the other constant. You can create perspective projections with this mechanism easily.

SHEAR = np.array([
    [1, -2],
    [0, 1]
    ])
a = np.dot(SHEAR, a)

Another hand

You can add a second hand to the clock. First, make the vector longer:

a = np.array([0, 2, 1, 1])

Add an extra line to the plotting code block:

ax.plot([0, a[2]], [0, a[3]], linewidth=7, color="red")

The matrices now are a bit more complicated - they are (4, 4) square matrices. For instance to rotate both hands independently:

alpha = radians(120)
beta = radians(10)

ROT = np.array([
    [ cos(alpha), sin(alpha), 0, 0],
    [-sin(alpha), cos(alpha), 0, 0],
    [0, 0, cos(beta), sin(beta)],
    [0, 0, -sin(beta), cos(beta)]
])
a = np.dot(ROT, a)