1. Computer Graphics (fall 2009)
School of Computer Science
University of Seoul

2. Chap 5: Viewing Classical and Computer Viewing Viewing with a Computer
Positioning of the Camera
Simple Projections
Projections in OpenGL
Hidden-Surface Removal
Interactive Mesh Displays
Parallel-Projection Matrices
Perspective-Projection Matrices
Projections and Shadows

3. 5.1 Classical and Computer Viewing

4. Basic Elements Object Projector Projection Plane
COP (Center of Projection) or DOP (Direction of Projection)
Principal faces/directions

5. Classical Viewing Orthographic projections Axonometric projections
Isometric, dimetric, trimetric
Oblique projections
Perspective projections
One-, two-, three-point perspectives

6. 1. Orthographic Projections
Projectors perpendicular to the projection plane
Multiview orthographic projection
(image courtesy of

7. 2. Axonometric Projections
Projectors orthogonal to the projection plane
Projection plane can have any orientation with respect to the object
Isometric, dimetric, trimetric
How many principal faces is the projection plane is placed symmetrically with respect to

8. 3. Oblique Projections Most general parallel views
Projector can have an arbitrary angle with the projection plane

9. 4. Perspective Viewing To achieve real-looking images
One-, two-, three-point perspectives
Number of vanishing points
How many of three principal directions in the object are parallel to the projection plane?

10. 5.2 Viewing with a Camera

11. OpenGL Viewing
All the classical viewing can be implemented  HOW??? (§5.9 & §5.10)
Default camera
Orthographic
Projection plane: “z=0”
At the origin pointing to –z-direction
Up direction: +y-direction
Viewing volume: cube with side 2  objects outside of this box cannot be rendered! (“clipping”)
Objects behind the camera can be rendered! (only for orthogonal projections)

12. 5.3 Positioning of the Camera

13. Positioning the Camera
No separate “viewing” transformation  concatenated into model-view matrices
Object coords.world coords.eye coords.
What matters is the relative position of objects and the camera
At any given time, the state of the model-view matrix encapsulates the relationship between the camera frame and the object frame

14. Positioning the Camera (cont’d)
Separate matrices For model & viewing
Model-view matrix (OpenGL)
glMatrixMode(GL_MODEL);
glMultMatrix(Mo);
render_object();
glMatrixMode(GL_VIEW);
glMultMatrix(Mc);
glMatrixMode(GL_MODELVIEW);
glMultMatrix(inv(Mc));
glMultMatrix(Mo);
render_object();

15. Viewing-Coordinate System
Defined by
Position (VRP: view-reference point, p)
Viewing direction (VPN: view-plane normal , n)
Up direction (VUP: view-up vector, vup)
Projected on the view plane (v)
3rd orthogonal direction (u) obtained by cross product
View-orientation matrix derivation? (advanced)
x, y, z axes  u, v, n
gluLookAt
Needs to be called before transforming objects
Error in the book (p.251)

16. Other Viewing APIs Roll, pitch, yaw Elevation, azimuth, twist
To specify the orientation
Ex) flight simulation
Elevation, azimuth, twist
Direction from the viewer
Ex) star in the sky

17. 5.4 Simple Projections

18. Perspective Projections
All projectors pass through the origin
(x,y,z)(xp,yp,zp)
Nonlinear, not affine, irreversible
Perspective division required
“Extended” homogeneous coordinates required  3D points = 4D lines (through the origin): (x,y,z,1)(x,y,z,z/d)=(xp,yp,zp,1)
How does the matrix look like?

19. Orthogonal Projections
No division required
(x,y,z)(x,y,0)=(xp,yp,zp)
How does the matrix look like?

20. 5.5 Projections in OpenGL

21. Projections in OpenGL Parameters in eye coordinates
Near/far clipping planes

22. Perspective Viewing in OpenGL
glFrustum
Need not be symmetric
gluPerspective
Symmetric
Defined using fov (field of view)  error in the figure 5.28 of the textbook
Calls glFrustum internally
near & far parameters must be positive
(image courtesy of the redbook)

23. Parallel Viewing in OpenGL
glOrtho
gluOrtho2D
No sign restrictions on near & far parameters

24. 5.6 Hidden-Surface Removal

25. Hidden-Surface Removal
How to determine which object is closer than others?
What happens without hidden-surface removal?
Two classes
Object space algorithm
BSP (Binary Space Partitioning) tree  restriction?
Image space algorithm
Z-buffer algorithm  used in (most) interactive graphics system including OpenGL
Complexity proportional to the resolution
Small overhead
More in Chap 7
Culling
Back faces of closed objects not rendered
Number of primitives reduced early

26. 5.7 Interactive Mesh Displays

27. 5.8 Parallel-Projection Matrices

28. Projection Normalization
All projections are converted into orthogonal projections by first distorting the objects such that the orthogonal projection of the distorted objects is the same as the desired projection of the original objects.
Simplifies clipping & hidden-surface removal (Chap 7)

29. Orthogonal-Projection Matrices
Normalization (by OpenGL projection matrix)
Converts the specified viewing volume to canonical view volume ([-1,1]x[-1,1]x[-1,1] cube)
Translation followed by scaling
Orthographic projection
(x,y,z)  (x,y,0)

30. Oblique Projections We can either Define the 4x4 matrix directly or
Implement by shear followed by orthographic projection (In practice, normalization is required in between)

31. 5.9 Perspective-Projection Matrices

32. Simple Perspective Projection
Frustum defined by x=±z, y=±z, z_max, z_min
For the perspective-normalization matrix N, what converts the planes as follows?
x=±z  x’’=±1
y=±z  y’’=±1
z=z_max  z’’=1
z=z_minz’’=-1

33. Perspective Projection
Depth ordering preserved by perspective-projection matrix  hidden-surface removal works in the normalized volume
General perspective projection
Apply shear to convert the asymmetric frustum to a symmetric one
Scale to the “simple” frustum

34. 5.10 Projections and Shadows

35. Shadows
A point is in shadow if it is not illuminated by any light source, or equivalently if a viewer at that point cannot see any light source.
Shadow polygon

36. Rendering Shadow Render twice: object polygon and shadow polygon
How to find the shadow polygon?  perspective projection (with light source as the camera)
Works only for shadows on flat surface
More on Chap 12