Page 90 - DCAP504_Computer Graphics

Unit 6: 2-D Transformation

The figure 6.10 depicts translation and rotation.

Figure 6.10: Translation and Rotation

As shown in figure 6.10, the square object is transformed using translation transformation. The
rectangle remains as a rectangle even though the position of the rectangle is changed. Similarly, as
shown in figure 6.10, when the rectangle is rotated using rotation transform, the angle of the lines with
respect to the origin varies, but the rectangle remains as a rectangle. Such transformations are called
affine transformation.
Affine transformation is useful while performing combination transformation. The combining of
rotation and scaling transformation can also be called as rotation-enlargement transformation. Consider
point coordinates P=(x, y).
When rotation-enlargement transformation is applied to this point, the transformation point P'=(x', y’).

  ' x cos α −sin α  0  x 
     

  ' y = s  sin α cos α  0  y
     
 ' z   0 0 1   z 
Where, ‘s’ is the scaling factor and α is the angle of rotation.
  ' x  cos.x α − . sin α 
y
   
y
  ' y = s  sin.x α + . cos α 
   
 ' z   z 
The equation is simplified as,
x’=s( cos. x α − . sin α )= . s( cosα x ) − . s ( sin α ) y
y
y’=s( sin.x α + . cos α ) = .s( sin α x ) + . s ( cosα ) y
y
z’=s (z)
Let,
a = cos.s α

b = - sin.s α
Then,
x’=ax + by
y’=ax - by

LOVELY PROFESSIONAL UNIVERSITY 83