The Parallelogram Image Package

The routine pargrm_image_2d allows you to readily view the geometry of a linear transformation of ${\bf R^2}$. After starting Maple, you need to load the package with a line like:

read `Pacific:othermaple:pargrm_image`;

where you will likely have to substitute for Pacific the name of your machine. The quotes here are backquotes in the upper left of your keyboard rather than apostrophes.

Suppose that vectors ${\bf v1}$ and ${\bf v2}$ are defined as in

v1 := vector([1,0]);
v2 := vector([0,1]);

and a matrix ${\bf A1}$ is defined by

A1 := matrix(2,2,[.28, .96, .96, -.28]);

Then the Maple call

gr1 := pargrm_image_2d([v1,v2],A1);
display(gr1, scaling=constrained):

will display two parallelograms with vertices at the origin. One parallelogram P1 will have edges ${\bf v1}$ and ${\bf v2}$, and be colored blue. The other will be the image under ${\bf A1}$ of this parallelogram, and so will have edges ${\bf A(v1)}$, and ${\bf A(v2)}$. This image is colored red.

At places where the original and image parallelograms overlap, the color will be blue, but you will still be able to see the edges ${\bf A(v1)}$ and ${\bf A(v2)}$.

The line scaling=constrained above tells Maple that you'd like the x and y axis scales to be the same. This is essential if you are trying to read off information like distance and angles from a picture !

There is also a routine pargrm_image_3d using 3 dimensional vectors, and a 3 x 3 matrix ${\bf A}$ to draw the images under ${\bf A}$ of a parallelepiped with edges ${\bf v1}$, ${\bf v2}$, and ${\bf v3}$. A typical call would be

pargrm_image_3d([v1,v2,v3],A);

Samples using the parallelogram image package are located in the file :Maple V Release 4:Math 221:Parallelogram Image on each Macintosh in the Lab.