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3D Perspective Projection (w)


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#1 adam17   Members   -  Reputation: 227

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Posted 07 October 2012 - 10:01 PM

I'm working on an assignment for my graphics class and I've encountered a problem. Basically I have a set of coordinates in World space and I have to transform them into Screen space and display them. I have to manually build all of the matrices and do the transformations by myself (no opengl or d3d). I have the following 3 matrices
  • Xsp (proj to screen transform)
  • Xpi (image to projection transform)
  • Xiw (world to image transform)
I'm rendering to a 256x256 frame buffer, so my Xsp matrix looks like this:
128	 0	 0	 128
0	 -128	 0	 128
0	 0	 1	 0
0	 0	 0	 1

My Xpi matrix has m[3][2] set to tan(fov / 2) where my fov = 35. (In the matrix .32 was just rounded for easier viewing)
1	 0	 0	 0	
0	 1	 0	 0
0	 0	 1	 0
0	 0	 .32	 0

Lastly my Xiw matrix is just the inverse of my camera transform. The camera is at 0,0,-15, looking at 0,0,0 with an up-vector of 0,1,0
1	 0	 0	 0	
0	 1	 0	 0
0	 0	 1	 15
0	 0	 0	 1

Using just these three matrices I can multiply them against a coordinate and get it transformed into screen space just fine. The catch is when I want them to be perspective correct. I've tried figuring out what 'w' is, but I cannot find a solid definition of how to calculate it. Some places are saying that the final Z value is used to divide X and Y by, but it doesn't look right. Some places are saying the W coord that is calculated by multiplying the matrix with the coordinate (assuming 1.0 for the 4th coord) is the W to divide by. I can't seem to get either method to work.

My question is what is W and how is it calculated?

Thank you so much in advance!
-Adam

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#2 Tournicoti   Prime Members   -  Reputation: 683

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Posted 07 October 2012 - 11:26 PM

Hello smile.png


With homogeneous coordinates, there are four components to represent a 3d vertex or a 3d vector :

  • if w != 0 , ( x , y , z , w ) represents the vertex ( x/w , y/w , z/w )
  • if w = 0 , it represents all the vectors colinear to the vector ( x , y , z ).   [ i.e. : a * ( x , y , z ) , a in lR*  ]

So for a vertex ( x , y , z ) transformed into ( x' , y' , z' ) with Transform :

  • ( X , Y , Z , W ) = ( x , y , z , 1 ) * Transform
  • ( x' , y' , z' ) = ( X , Y , Z ) / W

NB :
For the vertices, the w component is usually initially set to 1 so that the w division is not needed in rotations, translations, scalings, and any combination of them.
With this assumption, the w division is necessary only when Transform is (or 'contains') a projection matrix.

Nico


Edited by Tournicoti, 11 March 2013 - 03:45 AM.





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