Page 86 - DCAP504_Computer Graphics

Unit 6: 2-D Transformation

   4 
P’ = 8

 
 1 
Therefore the value of P’ in homogeneous representation is [4 8 1]. The point
coordinate (2, 4) after scaling by a factor=2 is (4, 8).

We have learnt in the beginning of the unit that translation transformation is performed
by adding translation matrix and point coordinate matrix. However, in homogeneous
coordinate system, the translation transformation is performed by multiplying the
transformation matrix and homogeneous representation of the object’s point coordinate
values.
6.3 Composite and Inverse Transformations

The linear transformations that are represented using matrices can be easily composed, i.e., combined. It
can also be inverted or reversed, i.e., to get the original object from the transformation. The use of
matrix and homogeneous representation helps to perform transformations such as translation, scaling,
rotation, reflection, and searing using a single matrix.
Composite Transformation
Now let us learn how to compose or combine two transformations? This can be achieved by simple
matrix multiplication. Consider A and B as the matrices of any two linear transformations which you
want to apply to a 2-D object. The effect of applying ‘A’ transformation first and then ‘B’ transformation
to the object can be achieved by multiplying A and B transformation, and then applying the resultant
transformation to the object.

Consider that the rectangle object with coordinate values represented in
homogeneous coordinate system representation are [0 0 1], [2 0 1], [2 2 1] and [0
2 1]. The rectangle has to be scaled by a scaling factor 0.5 in X and Y directions.
After scaling the matrix it has to be rotated by an angle 90 degree
counterclockwise about the origin.
The scaling matrix is given by,

 50. 0  0 

Scaling matrix=  0 0. 5  0
 
 0 0 1 
The coordinates of the rectangle after applying scaling transformation are [0 0
1], [1 0 1], [1 1 1], and [0 1 1].
Rotation matrix to obtain the rotation of 90 degree counterclockwise,

cos 90 − sin 90  0  0 −1  0 


=  sin 90 cos 90  0 = 1 0  0
   
 0 0 1   0 0 1 
The rotation transformation is applied to the scaled matrix. The coordinates of
the scaled rectangle after applying rotation transformation are [0 0 1], [0 1 1], [-1
1 1] and [-1 0 1].
Figure 6.9 shows all the three rectangle objects, i.e., original object, original
object after scaling transformation, and rotation of the scaled object.
The same result can be obtained by multiplying the scaling matrix and rotation
matrix and then multiplying the point coordinate matrix values of the rectangle

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