1. Viewing and Projection

2. Parallel Projection

3. Parallel Projections (known aliases): Orthographic or Isometric Projection

4. Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009
Oblique Projection
Arbitrary relationship between projectors and projection plane
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

5. Parallel Projections (known aliases): Oblique Projection L 

6. Perspective Projection
Projectors coverge at center of projection
Angel: Interactive Computer Graphics 5E © Addison-Wesley 2009

7. Perspective Projection
foreshortening - the farther an object is from the camera , the smaller it appears in the final image

8. Perspective Projection
The geometry of the situation is that of similar triangles. View from above:
What is x’ ?
P (x, y, z) X Z
View plane
(0,0,0)
x’ = ? d

9. Perspective Projection Top View
P=(xp,yp,zp)
P’=(x’,y’,z’)
CoP=(xc,yc,zc) xp x’ z’ zp

10. Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009
Simple Perspective
Consider a simple perspective with the COP at the origin, the near clipping plane at z = -1, and a 90 degree field of view determined by the planes
x = z, y = z
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

11. Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009
Perspective Matrices
Simple projection matrix in homogeneous coordinates
Note that this matrix is independent of the far clipping plane
M =
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

12. Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009
Generalization
N =
after perspective division, the point (x, y, z, 1) goes to
x’’ = x/z
y’’ = y/z
Z’’ = -(a+b/z)
which projects orthogonally to the desired point
regardless of a and b
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

13. What the Perspective Matrix means
Note: Normalized Device Coordinates are a LEFT-HANDED Coordinate system

14. Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009
Picking a and b
If we pick
a =
b =
the near plane is mapped to z = -1
the far plane is mapped to z =1
and the sides are mapped to x =  1, y =  1
Hence the new clipping volume is the default clipping volume
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

15. Normalization Transformation
distorted object
projects correctly
new clipping
volume
original object
original clipping
volume
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

16. Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009
OpenGL Perspective
glFrustum allows for an unsymmetric viewing frustum (although gluPerspective does not)
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

17. OpenGL Perspective Matrix
The normalization in glFrustum requires an initial shear to form a right viewing pyramid, followed by a scaling to get the normalized perspective volume. Finally, the perspective matrix results in needing only a final orthogonal transformation
P = NSH
our previously defined
perspective matrix
shear and scale
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

18. glFrustum Matrix

19. Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009
Why do we do it this way?
Normalization allows for a single pipeline for both perspective and orthogonal viewing
We stay in four dimensional homogeneous coordinates as long as possible to retain three-dimensional information needed for hidden-surface removal and shading
We simplify clipping
Angel: Interactive Computer Graphics 5 © Addison-Wesley 2009

20. What the Perspective Matrix means
Note: Normalized Device Coordinates are a LEFT-HANDED Coordinate system

21. Graphics Pipeline So Far
Object
Object Coordinates
Transformation
Object -> World
World
World Coordinates
Projection Xform
World -> Projection
Camera
Projection Coordinates
Normalize Xform & Clipping
Projection -> Normalized
Viewport
Normalized Coordinates
Viewport Transform
Normalized -> Device
Screen
Device Coordinates

22. What happens to an object...
Transformation
Object -> World
Object
Object Coordinates
World
World Coordinates

23. What happens to an object...
Transformation - Modelview
World -> Eye/Camera
World
World Coordinates
Viewport
Viewport Coordinates

24. What happens to an object...
Transformation - Projection
(Includes Perspective Divide)
Eye/Camera ->View Plane
Viewport
Viewport Coordinates
Rasterization
Scan Converting Triangles