1. Advanced Graphics &
Animation
3D Viewing Pipeline

2. Three-dimensional Viewing Pipeline
Transform into view coordinates
and Canonical view volume
Clip against canonical view
volume
Project on to view plane
Map into viewport
Transform to physical
Device coordinates
transform
clip
transform
World coordinates(3D)
View coordinates(3D)
View coordinates(3D)
View coordinates(2D)
Normalized device coordinates
Physical device coordinates

3. Parallel Projection
 Mostly used by drafters and engineers to create working drawings
of an object which preserves its scale and shape.
 The distance between the COP and the projection plane is infinite
i.e. The projectors are parallel to each other and have a fixed
direction.
P1
P2
P1‟
P2‟
Projection plane

4. • Orthographic projection:
here direction of projection is perpendicular to the
view plane.
• Axonometric projection:
when direction of projection is not parallel to any of
three principal axes.

5. Perspective Projection
• Generalization of the principles used by artists in drawing of scenes.
• It takes object representation in view space (L.H.S.) and produce a
projection on the view plane (canvas used by the artist).
• The projection of a 3D point onto the viewplane is the intersection of
the line from the point to the COP (eye position of the artist).

6. Cont.
• Distance between the COP and projection plane is finite.
• Perspective projection does not preserve object scale and shape.
P1
P2
P1‟
P2‟
Projection plane
COP

7. Perspective Anomalies
Perspective foreshortening
- the farther an object from the COP, the smaller it appears (i.e. its
projected size becomes smaller).
cop
View plane

8. Vanishing points
- there is an illusion that certain sets of parallel lines (that are not
parallel to the view plane ) appear to meet at some point on the view
plane.
- the vanishing point for any set of parallel lines that are parallel
to one of principal axis is referred to as a principal vanishing point
(PVP).
- the number of PVPs is determined by the number of principal
axes intersected by the view plane.

9. One principal vanishing point projection
- occurs when the projection plane is perpendicular to one of the
principal axes (x, y or z ).
Vanishing point
View plane is parallel to
XY-plane and intersects
Z-axis only.

10. Two principal vanishing point projection
X-axis
Vanishing point Z-axis
Vanishing point
View plane intersects
Both X and Z axis but
not the Y axis

11. • Three principal vanishing point intersection
View plane intersects all
Three of the principal axis
X, Y and Z axis
VP1
VP3
VP2

12. Deriving Perspective Projection
Assume
point vertex denoting COP : (xc,yc,zc)
point on the object : (x1,y1,z1)
representation of “projection ray” containing above two points
x = xc + ( x1-xc) u …………..eq 1
y = yc + ( y1-yc) u …………..eq 2
z = zc + ( z1-zc) u …………..eq 3
The projected point (x2, y2,D) will be the point where this line intersects the
XY plane . Putting z=0 for this intersection point in eq 3 .
u = - zc / z1-zc

13. Substituting into first two equations,
x2 =
y2 =
Value of D may be computed which is different from zero (to
preserve depth relationship between objects)
D = z1 / (z1 –zc)
z-z
zx-zx
c1
c11c
z-z
zy-zy
c1
c11c

14. Standard perspective projection
y
x
z
A (x,0,z)
A‟ (x‟,0,0)
P (x ,y, z)
P‟ (x‟,y‟,0)
C(0,0,-d)
do
(0,0,0)B (0,0,z)
z

15. Using similar triangles ABC and A‟OC,
x‟ = d.x / (z+d)
y‟ = d.y / (z+d)
z‟ = 0
matrix representation :
d100
0000
00d0
000d

16.  viewing based on synthetic camera analogy.
Specifying an arbitrary 3D view

17. By selecting different viewing parameters, user
can position the synthetic camera.
View reference
point
View-up vector
View plane

18. Effect of change of viewing parameters
 Imagine a string tied to „view reference point‟ on one end and to the
synthetic camera on the other end.
 By changing viewing parameters, we can swing the camera through
the arc or change the length of the string.
- changing the view distance is equivalent to how far away
from the object the camera is when it takes the picture.
- changing the view reference point will change the part of the
object that is shown at the origin.

19. Cont.
- changing the view plane normal is equivalent to
taking photograph of object from different orientations.
- changing view-up is equivalent to twisting the camera
in our hands. It fixes the camera angle.

20. View Volume
- The view volume bounds that portion of the
3D space that is to be clipped out and
projected onto the view plane.

21. View Volume for Perspective Projection
- its shape is semi-infinite pyramid with apex at the
view point and edge passing through the corners of the
window.
cop
View
window
Front clipping
plane
Back clipping
plane
Frustum view volume

22. View Volume for Parallel Projection
-It's shape is an infinite parallelepiped with sides parallel to the
direction of projection.
Parallelepiped
Viewed volume
View
window
Front clipping
plane
Back clipping
plane

23. Producing a Canonical view volume for a
perspective projection
cop
View
window
Front
clip
Back clip
View frustum
centerlineGeneral shape
for the Perspective
View volume
View volume

24. Step 1: shear the view volume so that centerline of the frustum is
perpendicular to the view plane and passes through the center of the
view window.
Frustum centerline
View volume

25. Step2: scale view volume inversely proportional to the distance from
the view window, so that shape of view volume becomes rectangular
parallelepiped.
View volume

26. Converting object coordinates to view plane
coordinates
 similar to the process of rotation about an arbitrary axis
zw
Yw
xw
World coordinate system
Yv
Xv
VRP
View plane (eye)
coordinate system

27. Steps:
1. Translate origin to view reference point (VRP).
2. Translate along the view plane normal by view distance.
3. Align object coordinate‟s z-axis with view plane coordinates z-
axis (the view plane normal).
a)- Rotate about x-axis to place the line (ie. Object coordinates
z-axis) in the view plane coordinates xz-plane.
b)- Rotate about y-axis to move the z axis to its proper position.
c)- Rotate about the z-axis until x and y axis are in place in the
view plane coordinates.

28. Ref:
„Computer Graphics‟ by S. Harrington (pp. 279-284)