Projections and Hidden
Surface Removal
Angel 5.4-5.6
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Objectives
• To discuss projection matrices to achieve desired perspective
• Discuss OpenGL API to achieve desired perspectives
• Explore hidden surface removal APIs in
OpenGL
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Projections and Normalization
• The default projection in the eye (camera) frame is orthogonal
• For points within the default view volume x p y p
= x
= y z p
= 0
• Most graphics systems use view normalization
– All other views are converted to the default view by transformations that determine the projection matrix
– Allows use of the same pipeline for all views
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Homogeneous Coordinate Representation default orthographic projection x p y p
= x
= y z p w p
= 0
= 1
M =

1

 0


0
 0 p p
= Mp
0
1
0
0
0
0
0
0
0
0
0
1






In practice, we can let M = I and set the z term to zero later
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Camera Setups
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Simple Perspective
• Center of projection at the origin
• Projection plane z = d , d < 0
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Perspective Equations
• Consider top and side views x p
= x z / d y p
= y z / d z p
= d
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Perspective Transformation
• Projection process
– Division by z
– Non-uniform foreshortening
• Farther from COP are smaller
• Closer to COP are larger
• Transform (x,y,z) to (x p
, y p
, z p
)
– Irreversible
– All points along projector results to same point
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Homogeneous Coordinates
• If w != 0 p
 x z y
1


















 wx wy wz w






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Homogeneous Coordinate Form consider q = Mp where M =






1
0
0
0
0
1
0
0 1 /
0
0
1 d
0
0
0
0





 p =





 x z y
1






 q =





 z / x y z d






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Perspective Division
• However w

1, so we must divide by w to return from homogeneous coordinates
• This perspective division yields x p
= x z / d y p
= y z / d z p
= d the desired perspective equations
• We will consider the corresponding clipping volume with the OpenGL functions
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Perspective Division (cont) q






 x p z y
1 p p














  z z x y d
1 d d







 
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Orthogonal Projection





 x p y z
1 p p













1
0
0
0
0
1
0
0
0
0
0
0
0
1



0

0

 x y z
1












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OpenGL Orthogonal Viewing glOrtho(left,right,bottom,top,near,far) near and far measured from camera
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OpenGL Perspective glFrustum(left,right,bottom,top,near,far)
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Using Field of View
• With glFrustum it is often difficult to get the desired view
• gluPerpective(fovy, aspect, near, far) often provides a better interface front plane aspect = w/h
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