**Illustration:** Find the normalization transformation N that uses the rectangle W (1, 1), X (5, 3), Y (4, 5) and Z (0, 3) as a window and also the normalized device screen like the viewport.

** **

**Figure: Example Transformations**

Currently, we observe that the window edges are not parallel to the coordinate axes. Consequently we will first rotate the window regarding W hence it is aligned along with the axes.

Now, tan α= (3 -1)/(5-1) = 1/2

⇒ Sin α = 1 /√5; Cos α = 2/√5

Now, we are rotating the rectangle in clockwise direction. Consequently α is negative which is, - α.

The rotation matrix about W (1, 1):

[T_{R.θ}]_{W} =

The x extent of the rotated window is the length of WX:

√(4^{2} + 2^{2}) = 2√5

As same, the y extent is length of WZ that is,

√ (1^{2} + 2^{2}) = √5

For scaling the rotated window to the normalized viewport we calculate s_{x} and s_{y} as,

s_{x} = (viewport x extent)/(window x extent)= 1/2√5

s_{y} = (viewport y extent)/(window y extent) = 1/√5

As in expression (1), the common form of transformation matrix showing mapping of a window to a viewport:

[T] =

Within this problem [T] may be termed as N as this is a case of normalization transformation with,

xw_{min} = 1 xv_{min} = 0

yw_{min} = 1 yv_{min} = 0

s_{x} = 1/2√5

s_{y} = 1/√5

Via substituting the above values in [T] which is N:

N =

Here, we compose the rotation and transformation N to determine the needed viewing transformation NR.

N_{R} = N [T_{R.θ}]_{W} =