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        Math 2270 sec. 3                          Laboratory Assignment  3.

        Treibergs                                 Due Oct. 19, 1998

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Create a document in which you answer the following questions, using a mixture 
of MAPLE computations and textual insertions (using "#" or text mode to comment
or usehandwritten text.) Print out a copy and hand it in. Remember to put your
name and section number on your paper. 

LINEAR TRANSFORMATIONS

This project will study the geometry of transformations. It will use two subrou-
tines which will help in visualizing transformations by animating repeated 
application of the transformation. 

A.)     Execute my MAPLE tutorial. You can type in the instructions or you can 
        download them from my account as follows: using NETSCAPE or whatever 
        web browser, open my home page at
                www.math.utah.edu/~treiberg
        Go to the M2270 home page and then click the Assignment 3 ANIMATION 
        ROUTINES. Save this file to your account and assign a file name, say  
        "AR.txt". Then launch
                xmaple AR.txt
        MAPLE will ask for type of input. Answer MAPLE TEXT. You should see a 
        copy the tutorial with input lines. Execute the instructions simply by 
        pushing your ENTER key. It may happen that MAPLE gets bogged down by 
        heavy memory demands because of the iterated graphics commands. In that 
        case simply restart the file, reenter the first few instructions (at 
        least through the definition of "letter:=") then skip down to where you
        left off and continue. Otherwise enter "restart" and reenter the first 
        few instructions. If the picture is broken, use your mouse to downsize 
        the picture to get better graphics. When handing in your graphics, down-
        size the pictures to speed up printout and to fit the pictures on the 
        page. To see the animations, use the mouse to activate the output 
        window. Then the buttons on the menu bar are movie controls, stop, go, 
        single step, loop etc.

B.)     By modifying the MAPLE TUTORIAL or by typing in the instructions your-
        self answer the following questions.

 1a.    Define  "letter" to be a list of points describing the figure you wish 
        to transform. Your figure may be my letter or anything else except the 
        unit square.
  b.    Define a transformation  f(x) = Ax  where  A  is the matrix

                       [  1      2 ]
                  A  = [           ]
                       [  3      4 ]
        
         Create a plot on which you show on the same plot the letter, the 
         vectors  i  and  j, and the  x  and  y  axes  before transformation and
         the transformed letter, the transformed vectors f(i)  and  f(j)   and 
         the transformed axes.

2.      For this problem, answer the question for ONE of the transformations: 
        dilation by a constant  c=2  or shear by a constant  k = -2. Create a 
        plot on which you show  the letter, the vectors  i  and  j, and the  x 
        and  y  axes  before transformation and the transformed letter, the 
        transformed vectors f(i)  and  f(j)   and the transformed axes.  Esti-
        mate the area of the letter before and after transformation.

3a.     Find a matrix  A  such that   f(x)  =  Ax is  reflection across the  
        y-axis.  Do 1b for this matrix.
 b.     Choose two nearby vectors such as  u=[-1,2]  and  v=[-3,1].  Let  R1  
        and  R2  denote reflections across the mirrors determined by  u  and  v.
        Plot the letter,  the letter after applying the reflection   R1, the 
        mirror line for  R1,  the letter after applying first R1 then R2, and 
        the mirror line for  R2  on the same plot.  Compare the letter, 
        R1(letter),  R2(R1(letter)) with regard to orientation and distortion. 
        Does  R2(R1(letter)) correspond to a rotation of the letter?

4.      A rigid motion is the combination possibly several reflections, rota-
        tions and translations in any order. Define  R  to be the rotations 
        matrix corresponding to rotation by an angel of  -Pi/6  and let  
        b=[0.1,-0.2]. Define the rigid motion f(x) = Rx + b.  Create a plot of 
        an animation of this transformation applied to your letter several 
        times. How would you describe rigid motions geometrically?