Viewing in 3D
1. Projections Types:
 perspective
 parallel / orthographic
2. Viewing Parameters:
 position of the viewer’s eye
 location of the viewing plane
 2-coordinate systems
o the scene (object)
o viewing eye coordinate system
3. Clipping:
 against 3D-view volume
4. Projection and displaying on viewport
Consider Projection Transformation :
Projection:
It is a process that transform points in a coordinate system of dimension n into points in a
coordinate system of dimension less than n.
Projection from 3D to 2D tools:
 projectors: straight projection rays
 center of projection (COP)
 projection plane ==> (not curved surface).
This type is called Planar Geometric Projection.
Types of Projections:
1. Perspective Projection
 projectors meet at center of projection (eye, camera)
 approximation of human visual system
 distortion close to COP
2. Parallel Projection
 Projectors are parallel
 COP is at infinity
1. Perspective Projection:
 The size of perspective projection of an object varies inversely with the distance
of that object from the center of projection
o not realistic
o distances cannot be taken from the projection
o angles are preserved on only those faces of the objects parallel to the
projection plane

Vanishing point:
o The perspective projection of any set of parallel lines that are not parallel
to projection plane converges to a vanishing point. ==> (parallel lines
meet at infinity ==> VP   )

Axis vanishing point:
o When the set of lines is parallel to one of the 3 axes(, x, y and z).
What is the maximum number of vanishing points?
One-point perspective projection of cube onto plane cutting the z-axis. The projection
plane normal is parallel to the z-axis.
2. Parallel Projections:
Depending on the relation between the direction of projection and the normal to the
projection plane:
a - orthographic :
both directions are the same (or the reverse of each other)
b - oblique:
direction of projection is not normal to the projection plane.
Examples:
(1) Orthographic Projection
(2) Oblique Projection
(projectors are not normal to projection plane)
(3) Isometric Projection:
It is the kind of projection generated when the projection plane normal makes equal
angles with each principle axis.
(4) Axonometric Orthographic Projection:
The type of projection that uses planes that are not normal to a principal axis and
therefore show several faces of the object at once.
Remarks on Parallel Projection:
 less realistic view
 can be used for exact measurements
 parallel lines remain parallel
 angles are preserved only on faces of the object parallel to the projection plane.
Projection of 3D scene onto 2D screen has two major components:
1 specification of a camera
2 specification of a viewing transformation
The viewing transformation:
specification of the parameters:
- A field-of-view angle, 
- Near and far bounding planes perpendicular to z.
- A 3D view of the camera and its viewing space as in the figure below:
Camera Viewing Space
Consider a side view of such a space:
Viewing Pyramid
note: u-axis coming out of the paper:
note:  forms a viewing volume in the shape of a pyramid with the camera at the apex of
the pyramid and the negative-w axis of the pyramid
The Viewing Transformation Matrix
Given the specs of parameters  , n, f  , we define the transformation of 3D scene
elements to the cube  1  u , v, w  1 is:
A ,n , f



0
0
0
cot( 2 )



 0
cot( )
0
0
2


f n
 0
0
 1


f n


2 fn
 0
0
0
f n


viewing transformation matrix
Development of the matrix:
- consider camera at origin
- use similar triangles:
d
v' 
v
w
d
u' 
u
w
The transformation that projects
du dv
du dv dw
(u , v, w)  (
,
,d)  (
,
,
)
w w
w w w
This can be expressed in H-D homogeneous coordinate:
(u, v, w,1)  (du, dv, dw, w)
In a Matrix form:
d 0 0
0 d 0
Pd  
0 0 d

0 0 0
0
0 
 1

0
Consider:
Need to transfer the viewing pyramid defined by , n and f into the cube  1  u , v, w  1 .
To transform the truncated viewing pyramid to the cube, P-matrix can be used:
d 0 0 0 
0 d 0 0 

P
(*)
 0 0 da  1


0 0 b 0 
where a and b are chosen constants which will cause the w values of the transformed
truncated viewing pyramid to lie in the range  1  w  1.
So we get:
(0,0,n) P  (0,0,1)
and
(0,0, f ) P  (0,0,1)
We have:
d 0 0 0 
0 d 0 0 
  0 0  dan  b n
0 0  n 1
 0 0 da  1


0 0 b 0 
and
d 0 0 0 
0 d 0 0 
  0 0  daf  b f 
0 0  f 1
 0 0 da  1


0 0 b 0 
Projecting back to the 3D we get:
 dan  b
(0,0,n) P  (0,0,
)
n
and
 daf  b
(0,0, f ) P  (0,0,
)
f
In order that the values on the left map to (0,0,1) and (0,0,-1) respectively we must have:
-dan+b = n
and
-daf+b = -f
subtract these equations and solve for (a)
daf  dan  f  n
2
f n
by substitution:
2 fn
b
f n
Inserting these values in our transformation matrix P indicated by * above
a
d 0
0 d

P  0 0


0 0

0
0
f n
f n
2 fn
f n
0
0 
 1


0
