General perspective transformation with cop at the origin, Computer Graphics

General Perspective transformation with COP at the origin

Here we suppose the given point P(x,y,z) be projected like P'(x',y',z') on the plane of projection. The center of projection is at the origin, determined by O(0,0,0). Let the plane of projection explained by the normal vector N=n1I+n2J+n3K and passing via the reference point R0(x0,y0,z0). By Figure 21, the vectors PO and P'O have the similar direction. The vector P'O is a factor of PO. Thus they are associated through the equation of: P'O = α PO, comparing elements we have x'=α.x   y'=α.y   z'=α.z we here get the value of α.

1015_General Perspective transformation with COP at the origin.png

We know about the equation of the projection plane passing via a reference point R0 and having a common vector as N=n1I+n2J+n3K is specified by PR0.N=0, which is:

(x-x0,y-y0,z-z0).( n1,n2,n3)=0 which is n1.( x-x0)+ n2.( y-y0)+ n3.( z-z0)=0 ---------( )

Because P'(x',y',z') lies upon this plane, hence we have as:

n1.( x'-x0)+ n2.( y'-y0)+ n3.( z'-z0)=0

Once substituting x'=α.x ;  y'=α.y ;  z'=α.z, we have as:

α =(n1.x0+ n2.y0+ n3.z0)/(n1.x+ n2.y+ n3.z) = d0/(n1.x+ n2.y+ n3.z)

This projection transformation cannot be shown as a 3x3 matrix transformation. Conversely, by utilizing the HC representation for 3-D, it can write in projection transformation as:

439_General Perspective transformation with COP at the origin 1.png

Hence, the projected point P'h(x',y',z',1) of given point Ph(x, y, z, 1) can be acquired as:


P'h = Ph. Pper,N, Ro = [x, y, z, 1]  

262_General Perspective transformation with COP at the origin 2.png

= [d0.x, d0.y, d0z, (n1.x + n2.y + n3.z)] ;

Here d0 = n1.x0 + n2.y0 + n3. z0.

Posted Date: 4/4/2013 3:26:10 AM | Location : United States

Related Discussions:- General perspective transformation with cop at the origin, Assignment Help, Ask Question on General perspective transformation with cop at the origin, Get Answer, Expert's Help, General perspective transformation with cop at the origin Discussions

Write discussion on General perspective transformation with cop at the origin
Your posts are moderated
Related Questions
Polygon representation methods - Space Partitioning Representations Space partitioning representations: this type of representation is used for explain the interior pr

Translation and shifting in Spatial Domain A) The three images shown below were blurred using square masks of sizes n=23, 25, and 45, respectively. The vertical bars on the le

Properties of Perspective projections - Transformation 1) Faraway objects seem smaller. 2) Straight lines are projected to straight lines. 3) Let line 1 and 2 is two s

Time Based and Presentation Tools In such authoring systems, components and events are organized beside a timeline, along with resolutions as high as 1/30 second. Time based to

Define Computer graphics.  Computer graphics remains one of the most popular and rapidly growing computer fields. Computer graphics may be explained as a pictorial representati

Projections When all display devices are 2D, you need to devise methods that give a realistic view of a 3D scene onto 2D display. With more and more devices coming in the marke

name some of the standard motion in key frames

Translation, Rotation and Scaling -  output primitives 1. Two basic approaches used in character generation are - Bitmap method and outline method.  2. All 2D geometric tran

Introduction of Curves and Surfaces This section has covered the methods of generating polygons, closed and curves surfaces. Under those methods we have discussed different ty

explainapplication of coherence in visible surface detection method