Page 88 - DCAP504_Computer Graphics

Unit 6: 2-D Transformation

Inverse Transformation
It is possible to obtain the original coordinate values of an object from its transformation. This is
achieved using inverse matrix. The matrix multiplication of the coordinate values of the transformed
object with the respective inverse transformation matrix, gives the original coordinate values of the
object.
Let us assume that P’ is a point coordinate of an object after transformation, T is the transformation
matrix, and P is the original coordinate of the object. The inverse of a transformation matrix T is T .
-1
Then, P= T .P’
-1
The inversion transformation matrices of various transformations are:

1 0 − dx  

Inverse translation matrix = 0 1 − dy 
 
 0 0 1 
 1 
 0  0
 Sx 

Inverse scaling matrix = 0 1 0 
 Sy 
 0 0 1 
 
 
 cos θ sin θ  0 

Inverse rotation matrix =  − sin θ cos θ  0
 
 0 0 1 

1 − a  0 

Inverse X-shear matrix= 0 1  0
 
 0 0 1 
 1 0  0 

Inverse Y-shear matrix=  − b 1  0
 
 0 0 1 

The homogeneous coordinates of a rectangle scaled by a scaling factor 0.5 in X
and Y directions are [0 0 1], [1 0 1], [1 1 1], and [0 1 1]. To find the homogeneous
coordinates of the original rectangle (i.e. before scaling), the homogeneous
coordinates of the scaled rectangle need to be individually multiplied with the
inverse scaling matrix.
Let the inverse scaling matrix with the scaling factor 0.5 in X and Y direction be
S,

 1 
 0  0
 5 . 0 
S =  0 1 0 
 5 . 0 
 0 0  1
 
 
2 0  0 

S =  0 2  0
 
 0 0 1 
By multiplying the homogeneous coordinates of the scaled rectangle with the

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