Page 85 - DCAP504_Computer Graphics

Computer Graphics

The matrix multiplication for 3X3 matrix is as show,
Let A and B are two 3X3 matrices,
a b  c 

A= d e  f
 
 g h i 
k l m  

B=  n o p 
 
 q r s 
 k.a + n + q l . a + o + r . a m + p +  s . c 
.
b
.
b
c
.
b
c
.
.

.
e
f
e
.
A*B=  k.d + n + q l . d + o + r . d m + p +  s . f
f
.
e
.
.
 
.
h
i
.
h
.
i
.
.
h
i
 k . g + . n + q l . g + o + r . g m + p + s 
The resultant matrix after multiplication is a 3X3 matrix.
If matrix A is 3X3 and B is 3X1 then the multiplication is carried out as shown,
a b  c 

A= d e  f
 
 g h i 
 x  

B=  y 
 
 z 
b
.
 x.a + y +  z . c 

A*B=  x.d + y + z 
.
e
.
f
 
h
.
i
 x . g + . y + z 
The resultant matrix after multiplication is a 3X1 matrix.
The 2-D point coordinate (2, 4) is scaled by a scaling factor 2 in X and Y
directions of the 2-D space.
To apply the transformation we need to obtain the homogeneous representation
of the point coordinates.
The homogeneous representation of (2, 4) is [2 4 1]. This is represented in matrix
form as,
   2 
P =  4 
 
 1 
2 0  0 

Scaling factor =  0 2  0
 
 0 0 1 
Let P’ be the matrix value after scaling,
P’ = Scaling factor * P
2 0   0    2 

= 0 2   0  4

   
 0 0 1   1 
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