Cleaning Up Coordinate Systems.

• Just a quick review.
• We start off by building an object.
  • I'm going to build a person. I like working on the 10x10 grid.
  • So I build a head
    • centered at (5,5)
    • 
  • I didn't communicate clearly with Bruce
  • so when he built the body
    • he built it centered at the origin
    • inside of the box (-3,-3,-3) to (3,3,3)
  • Mason points out that it would be best to have the entire thing in the box (-1,-1,-1) to (1,1,1)
    • So he scales the body by (2/3, 2/3, 2/3) $S_1$
    • And the head by (1/20,1/20,1/20) $S_2$
    • But the head is still needs to move the head to the right place, so he translates it by (-1/2, 1/6, 0) $T_1$
  • These are really three different sets of modeling coordinates.
  • 
  • $T_1S_2$ head
  • $S_1$ body
• Happy with this James decides to use our great model in his scene
  • He scales the entire thing by (20,20,20) $S_3$
  • And rotates it about the z axis by 90° $R_1$
  • And translates it to (40, 30, 0), $T_2$
  • This moves the figure to world coordinates.
  • $T_2R_1S_3T_1S_2$ head
  • $T_2R_1S_3S_1$ body
• We have one more world transformation
  • Alex decides to make the guy fly
  • So the entire scene is translated by $T_3$
  • This makes the final world transformation matrix
  • $T_3T_2R_1S_3T_1S_2$ head
  • $T_3T_2R_1S_3S_1$ body
• Mike wants to render this with a camera
  • placed at (0, 50, -30),
  • looking at (0,0,0)
  • with a up vector of (0,0,1). $C$
  • So we need to move the scene from world coordinates to the camera coordinates.
  • 
  • $C_1T_3T_2R_1S_3T_1S_2$ head
  • $C_1T_3T_2R_1S_3S_1$ body
• Finally, Michele defines the frustum as marked in green.
  • This will be $P_1$
  • It will both move the frustum to the default clipping volume for the vertex shader.
  • But after the vertex shader exits, it will project the image to normalized device coordinates (-1, -1) to (1,1)
  • 
  • $P_1C_1T_3T_2R_1S_3T_1S_2$ head
  • $P_1C_1T_3T_2R_1S_3S_1$ body
• Notice we have several parts to this
  • $P_1C_1$ happens to everything and only changes if
    • The camera changes
    • The frustum changes.
  • $T_3$ is sort of the final world transformation
    • This probably changes once per scene.
  • $T_2R_1S_3$ is the transformation on the entire "person"
    • This probably does not change.
  • Each part of the person has a transformation as well.
• So set the projection/camera transformation globally (send this over)
• Reset the world transformation (send this over)
• To draw each item in the world
  • Start with the top level transformation for the item.
  • Multiply in any modeling transformations (for each step if necessary)
  • When you get down to the object send this over
  • Draw the object.
• My code in the vertex shader might become

  • gl_Position = uniformProject * uniformTransform 
                      * uniformWorld * uniformModel *  attributePosition;
            

• We have one last piece to take care of in the transformation world.
• Normalized device coordinates to Device coordinates.
• The process we have discussed so far puts all vertexes in NDC space.
  • (-1,1) to (1,1).
  • But in general $(dx_{min}, dy_{min}), (dx_{max}, dy_{max}) $
  • I want to move the point to the unit cube.
  • So I need to scale by $(\frac{1}{dx_{max}-dx_{min}}, \frac{1}{dy_{max}-dy_{min}})$
  • And translate by $(-dx_{min},-dy_{min})$
• Finally I need to move this to DC, which will be
  • (0,0) to (width, height) in most cases
  • But remember, we can set the viewport.
  • But in general $(wx_{min}, wy_{min}), (wx_{max}, wy_{max}) $
  • So I need to scale by $(wx_{max}-wx_{min}, wy_{max}-wy_{min})$
  • And translate by $(wx_{min},wy_{min})$
• Fortunately this is done automatically by OpenGL.
• We might also need to reflect about the y axis (remember this from a long time ago)
• So there is one more thing, rotation in 3d.
  • But that needs to wait for another lecture.