The Journal of Online Mathematics and Its Applications

Volume 7. December 2007. Article ID 1677

Experiments with Matrices and Vectors in Calculus and Linear Algebra:
Using Mathematics to Transform and Animate Computer Images

Charles A. Sulewski, Gerald C. Kobylski, and Frank Wattenberg

Department of Mathematical Sciences

United States Military Academy

Abstract

This paper describes a Flash-based OSSLET (Open Source Shareable mathLET) that we have used at the United States Military Academy (USMA) with first year calculus students as a vehicle for motivating vectors, matrices, and linear and affine transformations and as an example of their power. Students use this OSSLET together with their knowledge of vectors and matrices to produce the kinds of computer-generated animations they see in movies like Superman, Star Wars, and Harry Potter. The paper includes all necessary files and instructions for students, instructors, and content developers along with a discussion of our experience with an earlier version of the OSSLET at USMA. As with all OSSLETs , the source code is included for those who are interested, although no programming is needed to use the OSSLET. The OSSLET may be freely used or modified without restriction.

Keywords

Technical Requirements

You will need two items to read and work with this article. Please follow the links to upgrade your browser, if necessary.

The imageAnimations OSSLET

Open the imageAnnimations OSSLET that is the focus of this article. The term OSSLET stands for Open Source Shareable mathLET. Please read the article Open Source Sharable Applet Collection for more background information.

You may resize the new window and arrange your desktop so that you can use this OSSLET as you read. Begin by clicking the green Play button. Notice both United States Military Academy (USMA) Black Knights spinning around the y-axis. This animation is produced by the one parameter family of linear transformations:

\left[{\begin{array}{c}y_1 \\ y_2 \\ \end{array}}\right] = \left[{\begin{array}{cc}\cos(4 \pi t) & 0\\ 0 & 1 \\ \end{array}}\right] \left[{\begin{array}{c}x_1 \\ x_2 \\ \end{array}}\right]

You can see the matrix in the upper left quadrant of the applet.

Now click the blue Jump Challenge item in the upper right corner and then click the green Play button again. Notice that only the left Black Knight moves and that the entries of the matrix have been reset to the identity matrix. This matrix produces the animation on the right -- namely the still animation. You and your students are challenged to reproduce the animation on the left by changing this matrix.

The OSSLET is very flexible. You can easily use your own demonstrations and questions and change the image in the animation. The OSSLET can produce animations using one parameter affine transformations of the form:

\vec y = A \vec x + \vec b

where

A = \left[{\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array}}\right] \quad \vec b = \left[{\begin{array}{c}b_1 \\ b_2 \\ \end{array}}\right] \quad \vec x = \left[{\begin{array}{c}x_1 \\ x_2 \\ \end{array}}\right] \quad \vec y = \left[{\begin{array}{c}y_1 \\ y_2 \\ \end{array}}\right]

These are the kinds of computer-generated animations our students see in movies like Star Wars and Harry Potter. We use this OSSLET at USMA to help students learn about vectors, matrices, and linear and affine transformations and to see some of the power of this mathematics. This article includes a discussion of this OSSLET and our experience using it. It also includes all the files you need to use this OSSLET. Although you do not need to know any programming to use the OSSLET, we do include the Flash source code for those who are interested. As part of the MAA OSSLET collection this OSSLET is freely available and may be used or modified without restriction.

Contents

Instructions

Experience using the imageAnimations OSSLET at USMA.

Thanks

This article was improved immeasurably by work at the NSF-funded, MAA sponsored workshop on Better Practices for Presenting Expository Mathematics on the Web held at the MAA Carriage House Conference Center, 15 - 21 July, 2007. Special thanks are due to Doug Ensley and Barbara Kaskosz for their wonderful introduction to Flash CS. Barbara has produced a wonderful parser which this OSSLET uses. Special thanks are also due to Troy Henderson who has lead the way in producing very high-quality graphics and text on the Web and to Kyle Siegrist who jump-started us on jsMath. David Cervone who produced jsMath also deserves great credit and thanks for a wonderful, easily used way to produce high quality resizable mathematical formulas on the web. We also want to thank our colleagues and cadets at USMA who contributed their enthusiasm and hard work to this project. We can't list them all by name but Jack Picciuto, Kerry Moores, Heather Jackson and Joe Lindquist deserve special thanks.