Transforming from Viewing Space to Normalized Device Coordinates

• A shot of the viewing transformation.
• 
• Stolen from this site
• The Red Book presents this view:
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• With more details on the step from (x,y,z,1.0) Object/Model coordinates to (x,y,z,w) homogeneous clip coordinates :
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• We have the first two boxes, but now we need the third:
• We generally model in a space like this:
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• After the transformation to eye (or camera) coordinates we have
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• Note, this can also be a view frustum, depending on the projection we decide to employ, but more on that in a bit.
• Switching to the new coordinate system requires a reflection about the xy plane.

• | 1  0  0  0 |
  | 0  1  0  0 |
  | 0  0 -1  0 |
  | 0  0  0  1 |
      

• Which will be added to the transformation matrices we derive in a bit.
• But we want our image to be in this type of space:
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• Moving to this space requires us to scale and reposition the image.

  • x-l    xndc-(-1)
    --- = --------
    r-l    1 -(-1)
    
    x-l    xndc+1
    --- = ------
    r-l     2
    
           2(x-l)
    xndc = ----- -1
            r-l
    
           2x - 2l -(r-l)
    xndc =  --------------
                r-l
    
           2x      r+l
    xndc = ----- - ---- 
           r-l     r-l
    
       

  • which becomes | 2x/r-l 0 0 -(r+l)/(r-l)| in a transformation matrix.
  • Similarly for y and z, except z is multiplied by -1

  • |  2x                r+l |
    | ---    0     0   - --- |
    | r-l                r-l |
    |                        |
    |         2y         t+b |
    |   0    ---   0   - --- |
    |        t-b         t-b |
    |                        |
    |             -2z    f+n |
    |   0     0   ---  - --- |
    |             f-n    f-n |
    |                        |
    |   0     0    0      1  |
    	 

  • This is an orthographic projection.
    • In this case, lines are projected orthogonal to the viewing plane.
    • To do this, the left, right, top, bottom and near and far distances need to be specified.
    • By default, in OpenGL, the values are -1, 1, -1, 1, -1, 1
    • The mat class includes an Ortho method for matrices.
• A prospective projection will look more "natural" to our eyes.
  • here is where we create view frustum.
  • 
  • This is specified with a left, right, top, bottom, near, far as before.
  • but in this case, lines all project to the eye or to the center of projection.
  • Frustum(left, right, bottom, top, near, far);