Find the normalization transformation, Computer Graphics

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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.

2190_Find the normalization transformation 1.png

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):

550_Find the normalization transformation 2.png

[TR.θ]W =

945_Find the normalization transformation 3.png

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

√(42 + 22) = 2√5

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

√ (12 + 22) =   √5

For scaling the rotated window to the normalized viewport we calculate sx and sy as,

 sx = (viewport x extent)/(window x extent)= 1/2√5

sy = (viewport y  extent)/(window y extent) =   1/√5

925_Find the normalization transformation 4.png

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,

xwmin = 1                        xvmin = 0

ywmin = 1                        yvmin = 0

 sx = 1/2√5      

 sy =  1/√5

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

N =

1677_Find the normalization transformation 5.png

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

 NR = N [TR.θ]W =

2096_Find the normalization transformation 6.png

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