1. Projections and Hidden Surface Removal
Angel
Angel: Interactive Computer Graphics5E © Addison-Wesley 2009

2. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Objectives
To discuss projection matrices to achieve desired perspective
Discuss OpenGL API to achieve desired perspectives
Explore hidden surface removal APIs in OpenGL
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

3. Projections and Normalization
The default projection in the eye (camera) frame is orthogonal
For points within the default view volume
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
xp = x
yp = y
zp = 0
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

4. Homogeneous Coordinate Representation
default orthographic projection
xp = x
yp = y
zp = 0
wp = 1
pp = Mp
M =
In practice, we can let M = I and set
the z term to zero later
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

5. Angel: Interactive Computer Graphics5E © Addison-Wesley 2009
Camera Setups
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6. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Simple Perspective
Center of projection at the origin
Projection plane z = d, d < 0
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

7. Perspective Equations
Consider top and side views
xp =
yp =
zp = d
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

8. 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 (xp, yp, zp)
Irreversible
All points along projector results to same point
Angel: Interactive Computer Graphics5E © Addison-Wesley 2009

9. Homogeneous Coordinates
If w != 0
Angel: Interactive Computer Graphics5E © Addison-Wesley 2009

10. Homogeneous Coordinate Form
consider q = Mp where
M =
p =
 q =
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

11. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Perspective Division
However w  1, so we must divide by w to return from homogeneous coordinates
This perspective division yields
the desired perspective equations
We will consider the corresponding clipping volume with the OpenGL functions
xp =
yp =
zp = d
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

12. Perspective Division (cont)
Angel: Interactive Computer Graphics5E © Addison-Wesley 2009

13. Orthogonal Projection
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14. OpenGL Orthogonal Viewing
glOrtho(left,right,bottom,top,near,far)
near and far measured from camera
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

15. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
OpenGL Perspective
glFrustum(left,right,bottom,top,near,far)
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

16. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
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
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009