MA 310				Lab Worksheet

Manipulating Graphs with Matrices in 2-Space

	There are many ways in which graphs can be manipulated: a graph can be reduced or enlarged in size, reflected about a line, sheared, or rotated. All of these operations can be performed by applying a matrix to the coordinates of a graph. These matrices are called transformation matrices. In this lab, you will write some m-files in Matlab to transform a graph and animate an image with a movie.

 1. First, we need a graph. We'll take a simple one like the letter A. As you read through this, you should start writing  m-files to carry out the plots. In what follows, lines starting with >> are lines for you to type on the command line, but don't type the >>.
    The letter A  can be drawn by connecting a set of vertices. Enter the following in Matlab:
>>x=[0,2,3,5,4.25,3.5,1.4,1.75,3.25,2.5,1.75,1.4,.75,0] ; 
>>y=[0,3,3,0,0,1,1,1.5,1.5,2.5,1.5,1,0,0] ;
Create the matrix M=[x;y], a matrix of 2 rows, 14 columns. In what follows, the x coordinates of the letter A are in row 1 of M. You can see these by typing M(1,:), which is row 1, and every column. The y-coordinates are M(2,:).

   To plot the letter in Matlab, you can use the command
>>plot(M(1,:),M(:,2));
   Or, to draw the letter in Matlab and color it red, use the command 
>> fill(M(1,:), M(2,:),'r');
>> hold on;   % Keep the graph so more plots can be applied
   If you like, you can change the axes by adding something like
>> axis([-1 6 -1 5]);  % this is the window [minx maxx miny maxy]

Dilation and Contraction Transformations
  To reduce the size of letter A, suppose we want to reduce the x direction by 0.5, and the y direction by 0.8.  Then the 2x2 transformation matrix T1=[0.5,0; 0,0.8] will provide the reduction.
Check some entries of W1 here to see how this matrix changes the x and y values in M.
>> W1=T1*M;
>> fill(W1(1,:),W1(2,:),'g');  % Draw the reduced A in green.
     Note here that W1(1,:) gives the x-coordinates in row1 of W1, and W1(2,:) gives the y-coordinates in W1.
  
Shear Transformations
  The letter A can also be sheared, something that is used to express fonts in italics. A shearing transformation can be applied to obtain many different results. In this example, we'll shear the letter A by  30 ˚to the right.  We'll start a new figure. Enter the following in Matlab.
>>figure;
>>fill(M(1,:), M(2,:),'r');
>> hold on;
>> theta=pi/6;
>> T2=[1,cos(theta); 0,1];
>> W2=T2*M;
>> fill(W2(1,:),W2(2,:),'b');

   To see what is happening, manually multiply matrix T2 by the first few columns in M to see how T2 changes the x-coordinates, but does not change the y-coordinates. You can change T2 to see how to perform other shears. 

Translation Transformations 
  To translate an image horizontally or vertically, we only need to add a constant to the x-coordinates and/or y-coordinates. For example, to move the letter A one unit to the right and up 2 units, we need a matrix of 14 columns of 1's and 14 columns of 2's. Enter the following in Matlab:
>>figure;
>>fill(M(1,:), M(2,:),'r');
>>axis([-2 6 -2 6]);
>> hold on;
>>L=[ones(1,14);2*ones(1,14)];
>> W3=M+L;  % shift M 1 unit to right, 2 units up.
>> fill(W3(1,:),W(2,:),'g');

 Rotation Transformations  
   An image like letter A can be rotated in many ways; for example about a point or line. See your calculus textbook for a derivation of a rotation matrix. For simplicity, we'll apply a rotation about the origin. 
>>figure;
>>fill(M(1,:), M(2,:),'r');
>> hold on;
>> theta=pi/6;
% The rotation matrix is
>>T3=[cos(theta), -sin(theta); sin(theta), cos(theta)];
>>W3=T3*M;
>>fill(W3(1,:),W3(2,:),'c');
% Let's make one more rotation
>>theta=pi;
>>T3=[cos(theta), -sin(theta); sin(theta), cos(theta)];
>>W3=T3*M;
>>fill(W3(1,:),W3(2,:),'c');

   We can continue with a series of rotations to create a rotation movie.
 Open a new m-file in Matlab and copy the following code. 

function []=movie2;
% Make a movie of rotating the letter A about the origin.
% Make up the x,y coordinates of the letter A, or any other figure
x=[0,2,3,5,4.25,3.5,1.4,1.75,3.25,2.5,1.75,1.4,.75,0];
y=[0,3,3,0,0,1,1,1.5,1.5,2.5,1.5,1,0,0];
M=[x;y];
fig=figure;
axis([-8 8 -8 8]);
set(gca,'nextplot','replacechildren');
mov=avifile('myname.avi','fps',10,'compression','None');
mov.quality=100;
% Loop through and create 24 frames to make the movie
for k=1:24
    theta=k*pi/6;
    T=[cos(theta), -sin(theta); sin(theta), cos(theta)];
    W=T*M;
    fill(W(1,:),W(2,:),'b');
    axis([-8 8 -8 8]);
    F(k)=getframe(fig);
    mov=addframe(mov,F(k));   
end;
mov=close(mov); % save your movie myname.avi to a folder
disp('movie created');
clf;
disp('playing movie..Press Ctrl C to stop');
%mov=aviread('myname.avi');
movie(F,5,10) % play the movie 5 time at 10 FPS