1. Viewing Transformations
Viewing parameters
Viewing projections
- Parallel
- Parallel orthographic projection
- Parallel oblique projection

2. Viewing transformations
Viewing transformations are concerned with projecting 3D objects onto 2D surfaces

3. Virtual camera Yw Zw View plane View up-direction V U N Viewing
distance Xw
Direction of gaze
Handedness axis
View Reference Point
(eye position)

4. Viewing projections Parallel centre of projection in the infinity
the projection lines are parallel X Y Z
Projection plane
Projection lines

5. Parallel orthographic projection
Projection lines perpendicular to the projection plane
Transformation matrix
xp = x
yp = y

6. Parallel oblique projection
Projection lines parallel, but not perpendicular to the projection plane
Projection lines
Projection plane Y X Z

7. Parallel oblique projectionX Y Z
P( 0,0 1) P' L a
L cos
L sin
Matrix

8. Parallel projections preserve relative dimensions of the objects
do not give realistic appearance

9. Perspective projections
The projection lines meet in one or more projection centre(s) Y X
COP
(Centre of Projection) Z

10. COP at the centre of the viewing coordinate system
projection plane is at distance D from the COP
on the positive part of the z axis (ZCOP = 0) y P P'
COP z D

11. P = (x, y, z)
P' = (xp, yp, D )

12. Transformation matrix
Example point transformation

13. Projection plane at the centre of the viewing coordinate system
projection plane is at distance D from the COP
on the negative part of the z axis (ZCOP = -D) y P P'
COP z D

14. z y D
COP P P'
P = (x, y, z)
P' = (xp, yp, 0 )

15. Transformation matrix
Example point transformation

16. Perspective projections
do not preserve relative dimensions
give realistic appearance

17. Clipping Used to restrict the depth of the projected scene
Back clipping plane
Front clipping plane
View plane