Mathematical description of a perspective projection, Computer Graphics

Mathematical description of a Perspective Projection

A perspective transformation is found by prescribing a center of projection and a viewing plane. Let here assume that P(x,y,z) be any object point in 3-D and center of projection is at E(0,0,-d). The problem is to find out the image point coordinates P'(x',y',z') on the Z=0 plane as in Figure 18.

703_Mathematical description of a Perspective Projection.png

The parametric equation of a l EP, beginning from E and passing via P is:

E+ t(P-E)  0

=(0,0,-d)+t[(x,y,z)-(0,0,-d)]

=(0,0,-d)+t(x,y,z+d)

=[t.x, t.y, -d+t.(z+d)]

Point P' is acquired, when t=t*

There is, P'=(x',y',z') =[t*.x, t*.y, -d+t*.(z+d)]

Because P' lies on Z=0 plane implies -d+t*.(z+d)=0 must be true, there is t*=d/(z+d) is actual.

Therefore x'=t*.x=x.d/(z+d)

                  y'=t*.y=y.d/(z+d)

                  z'=-d+t*(z+d)=0,

 

Hence P'=( x.d/(z+d), y.d/(z+d), 0)

                  =(x/((z/d)+1),y/((z/d)+1),0)

In terms of Homogeneous coordinate system here P'=(x,y,0,(z/d)+1).  ---------(5)

 

The equation 5 in above can be written in matrix form as:

21_Mathematical description of a Perspective Projection 1.png

-------(1)

There is, P'h = Ph.Pper,z   ------    (2)

Here Pper,z in equation (4) represents the single point perspective transformation on z-axis.

The Ordinary coordinates are as:

[x',y',z',1]=[x/(r.z+1),y/(r.z+1),0,1]  where r=1/d                             ------ (3)

Posted Date: 4/4/2013 2:50:14 AM | Location : United States







Related Discussions:- Mathematical description of a perspective projection, Assignment Help, Ask Question on Mathematical description of a perspective projection, Get Answer, Expert's Help, Mathematical description of a perspective projection Discussions

Write discussion on Mathematical description of a perspective projection
Your posts are moderated
Related Questions
Question 1: (a) Describe the term Mask Path and give brief steps how you could change a rectangle into a triangle with respect to time in AE CS3. (b) Expressions are ve

Problem with Interpolated Shading There are several more shading models that intermediate in complication among Gouraud and Phong shading, linking the liner interpolation of t

QUESTION 1. Describe two methods for converting 16x9 widescreen movie to 4x3. 2. Explain clearly the difference between File Format and Codec. 3. What is a video transiti

Where the video controller is used?  A special purpose processor, which is used to control the operation of the display device, is called as display controller or video control

Explain Various techniques of Character Generation Algorithm?

Sequence of Steps to Generate a Full Animation The sequence of steps to generate a full animation would be as given below: 1)  Improve a script or story for the animation.

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

slider crank

Application of animation: There are a variety of utilizations for computer animation. They can vary from fun to practical and educational ones. Medicine, military, entertainment,

what is fixed point scaling? how composit transformation techniques works on it