Understanding Transformation Matrix

July 4, 2017

In last few posts we had looked into various Affine transformation and we learned about the transformation matrix. Let’s summarize the Affine transformation matrices in this post with animation, which is written in pure Scilab script.

Run the following script in Scilab to see the animation:

// 4 points square
x = [0 1 1 0]';
y = [0 0 1 1]';
 
// Plot the points
plot(x,y,'*');
ax = gca();
ax.data_bounds = [-1 -1;5 5];
ax.children.children.mark_size = 10;
xnumb(x,y,1:4);
 
// Initialize points for the transformation
plot(x,y,'or');
e = gce();
ax = gca();
ax.children.children.mark_size = 10;
 
// Change this for slower animation, in ms
delay = 1;
cnt2 = 1;
// Translation
for cnt = [0:0.1:3 3:-0.1:0]   
    sleep(delay);
    m = [1 0; 0 1; cnt cnt];
    xy= [x y] * m(1:2,1:2) + repmat(m(3,:),4,1)
    e.children.data = xy;
    title('$ Translation\\ \left[\begin{array}{c}1 &amp; 0 \\0 &amp; 1 \\' + string(cnt) + '&amp;' + string(cnt) + '\end{array}\right] $','position',[-1 2],'fontsize',5);
end 
 
// Rotation
for cnt = [5:5:45 45:-5:0]     
    sleep(delay);
    m = [cosd(cnt) -sind(cnt); sind(cnt) cosd(cnt); 0 0];
    xy= [x y] * m(1:2,1:2) + repmat(m(3,:),4,1)
    e.children.data = xy;
    title('$ Rotation\\ \left[\begin{array}{c}cosd('+string(cnt)+') &amp; -sind('+string(cnt)+')\\-sind('+string(cnt)+') &amp; cosd('+string(cnt)+')\\ 0 &amp; 0\end{array}\right] $','position',[-1 2],'fontsize',5);
end 
 
// Scaling
for cnt = [1:0.1:5 5:-0.1:1]    
    sleep(delay);
    m = [cnt 0 ;0 cnt;0 0];
    xy= [x y] * m(1:2,1:2) + repmat(m(3,:),4,1)
    e.children.data = xy;
    title('$ Scaling\\ \left[\begin{array}{c}' + string(cnt) + ' &amp; 0 \\0 &amp; ' + string(cnt) + ' \\0 &amp; 0\end{array}\right] $','position',[-1 2],'fontsize',5);
 
end 
// Shearing
for cnt = [0:0.1:1 1:-0.1:0]    
    sleep(delay);
    m = [1 cnt ;cnt 1;0 0];
    xy= [x y] * m(1:2,1:2) + repmat(m(3,:),4,1)
    e.children.data = xy;
    title('$ Shearing\\ \left[\begin{array}{c}1 &amp; ' + string(cnt) + ' \\' + string(cnt) + ' &amp; 1 \\0 &amp; 0\end{array}\right] $','position',[-1 2],'fontsize',5);    
end

The following animation is the similar result you will see when you run the above script:

In summary, the Affine transform will reallocate the pixel value in the “blue star” location to the “red circle” location.

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