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 Three Dimensional Transformations A 3D geometric transformation is utilized extensively in object modelling and rendering. 2D transformations are naturally extended to

Aspect ratio - Display Devices Ratio of vertical points to horizontal points necessary to produce equal length lines in both directions on the screen. For example, in a CRT mon

1. Modify the DDA algorithm for negative sloped lines; discuss both the cases i.e., slope > 1 and 0   Ans. For the generation of lines along with negative slopes as:

Chemistry: Computer animation is a very helpful tool in chemistry. Several things in chemistry are too small to see, and handle or do experiments on like, molecules and atoms for

Distinguish between parallel and perspective projection Parallel Projection Perspective projection Coordinate position are transformed

Question (a) Define the term Multimedia. (b) Describe any four important tools you know about for a virtual campus. (c) Following our discussion in our lecture list an

Acquire the perspective transformation onto z = - 2 Plane, where (0, 0, 18) is the center of projection. Solution: Now centre of projection, C (a, b, c) = (0, 0, 18) ∴ (n 1

Ask question #Minimum how can we use stroke method method for character generation? 100 words accepted#

DV Encoder Type 2: Encoder Type 2 generates a VfW compatible AVI file format. Such file has separate streams for audio and video and it can also be processed through DirectShow. T