Find the normalization transformation, Computer Graphics

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

Posted Date: 4/3/2013 4:05:30 AM | Location : United States

Related Discussions:- Find the normalization transformation, Assignment Help, Ask Question on Find the normalization transformation, Get Answer, Expert's Help, Find the normalization transformation Discussions

Write discussion on Find the normalization transformation
Your posts are moderated
Related Questions
Explain the differences among a general graphics system designed for a programmer and one designed for a specific application, such as architectural design? Basically, package

Geometric Continuity There is another notion of continuity called geometric continuity. Although the idea existed in differential geometry, the concept was introduced for geome

What is the major disadvantages of storing information structures in display memory? A major disadvantage of storing information structures in display memory is that when color an

Basic Transformations - 2-d and 3-d Transformations Consider the xy-coordinate system upon a plane. An object or Obj in a plane can be identified as a set of points. All objec

Ellipse Generation Algorithm  You know that a circle is symmetric in all the octants, while ellipse is symmetric with respect to four quadrants.  Therefore, to draw an ellipse,

are you a robot

Explain window to view port transformation

What are the different ways of specifying spline curve?  Using a set of boundary conditions that are imposed on the spline. Using the state matrix that characteristics