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







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