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.
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:
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]
= [d0.x, d0.y, d0z, (n1.x + n2.y + n3.z)] ;
Here d0 = n1.x0 + n2.y0 + n3. z0.