Sample Curriculum Materials, Part II

Open the rotation figure that will be useful as we discuss how to rotate an image. The key idea is that the first column of the matrix A determines the location on the screen of the point in the image at (1, 0) and the second column determines the location on the screen of the point (0, 1) in the image since

\left[{\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array}}\right] \left[{\begin{array}{c}1 \\ 0 \\ \end{array}}\right] = \left[{\begin{array}{c} a_{11} \\ a_{21} \\ \end{array}}\right] \quad \quad {\rm and} \quad \quad \left[{\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array}}\right] \left[{\begin{array}{c} 0 \\ 1 \\ \end{array}}\right] = \left[{\begin{array}{c} a_{12} \\ a_{22} \\ \end{array}}\right].

By looking at the shaded triangle in the figure you just opened we see that after a counterclockwise rotation by an angle \theta the location on the screen of the point (1, 0) in the image is (\cos \theta, \sin \theta) or in column vector notation

\left[{\begin{array}{c} \cos \theta \\ \sin \theta \\ \end{array}}\right].

and by looking at the cross-hatched triangle we see that after a counterclockwise rotation by an angle \theta the location on the screen of the point (0, 1) in the image is (-\sin \theta, \cos \theta) or in column vector notation

\left[{\begin{array}{c} -\sin \theta \\ \cos \theta \\ \end{array}}\right].

A = \left[{\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{array}}\right]

In particular, to rotate the Black Knight counter-clockwise by 30^{\circ} or \pi/6 radians, we use the matrix

A = \left[{\begin{array}{cc} \cos (\pi/6) & -\sin (\pi/6) \\ \sin (\pi/6) & \cos (\pi/6) \\ \end{array}}\right]

Questions

For this set of questions you will need the same OSSLET that you used in the previous page; if you closed it, please re-open it now. Click each of the questions buttons -- Question 1 -- Question 7 -- in the upper right area of the window and reproduce the given image manipulation by changing the elements of the matrix A. Note these entries are highlighted in pink. To see each question and check your work, click the Play button.

Animations

Open the new OSSLET that we will use as we discuss animations. Click the Play button to see our first example. Notice the Black Knights spinning around. This effect is produced by using the matrix:

A = \left[{\begin{array}{cc} \cos (2\pi t) & -\sin (2\pi t) \\ \sin (2\pi t) & \cos (2\pi t) \\ \end{array}}\right]

When you click the Play button the parameter t is set to zero and the image is placed on the screen; then the parameter t is increased by a small amount and the image is placed on the screen again. This process is repeated many times until the value of the parameter t reaches 1 at which time the animation stops. This produces a series of rapidly changing still "frames" that produce the illusion of motion. This is exactly what is done in movies and on television.

Questions

For this set of questions use the same live window you used for the example above. Click each of the questions buttons -- Question 1 -- Question 6 -- in the upper right area of the window and reproduce the given animation by changing the elements of the matrix A. Note these entries are highlighted in pink. To see each question and check your work, click the Play button.