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
GRAPHICS: It is one of the core elements of any multimedia application. We all have heard a well-known saying as "one picture conveys a message of 1000 words", hence without

Phong Specular Reflection Model or Specular Reflection This model of local illumination is frequently termed as the Phong specular reflection model. Here we discuss the matter

What is the use of Projection reference point?  In Perspective projection, the object positions are transformed to the view plane with these converged projection line and the p

Computations with Phong Shading Computations involved along with Phong Shading:  i)   Find out average unit normal vector at each polygon vertex. ii)   Linearly interpol

Background Texturing is like wallpapering; you are pasting an image onto the OpenGL Quad primitive.  Recall that GL_QUAD is specified by four vertices.  An image, or a texture,

what type of animation is produced by the line y=mx+c?

Bezier curves: Theses are special polynomial curves expressed using Bernstein polynomials. Spline curves are simple extensions of Bézier curves composed of two or more polynom

Categories of Parallel Projection Parallel projection can be categorized as per to the angle which the direction of projection makes along with the projection plane. For illu

Software for computer animation Whether you might have the excellent hardware in the world, but without a high-quality software package, your hardware can act nothing. There

1. For the polygon shown in Figure on the next page, how many times will the vertex V 1 appear in the set of intersection points for the scan line passing through that point?  How