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
refreh buffer


Raster Scan Display Device - types of refresh monitors Now day screen display is also based on Cathode ray Tube technology, except that in place of displaying the picture tra

Find the normalization transformation N, which uses the rectangle W(1, 1); X(5, 3); Y(4, 5) and Z(0, 3) as a window and the normalized deice screen as viewpoint.

Pen Tablet and Plotter Pen Tablet: It is a digitizer tablet which is specialized for handwriting and hand marking. LCD-based tablets imitate the flow of ink as the tip tou

Question: (a) There are 3 main industries of drawing style and it has become a phenomenon. It was inspired by internationally diffused cartoons, comic strip books and since the

Illustration 1: How does the z-buffer algorithm find out which surfaces are hidden? Solution : Depth or Z-buffer algorithm employs a two buffer area each of two-dimensional ar

What are the important properties of Bezier Curve?  It requires only four control points It always passes by the first and last control points The curve lies enti

Polygon Clipping - Raster Graphics and Clipping After considerate the idea of line clipping and its algorithms, we can currently extend the idea of line clipping to polygon cl

Explain depth buffer method