1. Chap 3 Viewing Pipeline Reading:
Angel’s “Interactive Computer Graphics”, Sixth ed.
Sections 4.1~4.7
Chap 3 View Pipeline, Comp. Graphics (U) 1

2. Outline View parameters setting Projection in OpenGL
Simple perspective/parallel projection
Perspective-projection matrix
View volume clipping
Chap 3, View Pipeline (CG-U) 2

3. Viewing Pipeline Review…..
projection
View transform
View-Volume
Clipping in
homogeneous
space
Window-viewport
mapping
Chap 3, View Pipeline (CG-U) 3

4. A Practical Viewing System Review…..
Users specify
Parameters for view coordinate system
A camera position C
a viewing direction vector N
an upvector V.
View volume information
Projection type
A view plane distance d (= near clipping plane distance), and a far clipping plane distance f.
A view window on the view plane
Chap 3, View Pipeline (CG-U) 4

5. Viewing Parameters Review…..
View coordinate system -n
World coordinate system
Here, without near and far clipping planes
Chap 3, View Pipeline (CG-U) 5

6. OpenGL Setting for viewing system -1
In OpenGL, gluLookAt() function alters the model-view matrix
gluLookAt(eyex,eyey,eyez,atx,aty,atz,upx,upy,upz)
By default
a camera at the origin of the word frame pointing in the -z direction
the model-view matrix is initially an identity matrix.
Chap 3, View Pipeline (CG-U) 6

7. OpenGL Setting for viewing system -2
Look-at function
Eyepoint eye, specified in the world frame, determines VRP
At point at, specified in the world frame
Up vector up
gluLookAt(eyex,eyey,eyez,atx,aty,atz,upx,upy,upz)
Defines the model-view matrix
glMatrixMode(GL_MODELVIEW)
glLoadIdentity();
Chap 3, View Pipeline (CG-U) 7

8. OpenGL Setting for viewing system -3
Derive (u,v,n) defined in world coor. sys.
n = eye - at
Negative of viewing direction N
v: view up vector up
if up is perpendicular to n
u = v x n
up may not be perpendicular to n
v = up - its projection in direction of n, or
First do u = up x n, then v=n x u
Chap 3, View Pipeline (CG-U) 8

9. OpenGL Setting for viewing system -4
Chap 3, View Pipeline (CG-U) 9

10. OpenGL Specifying projection information
Properties of camera – focal length of lens or the size of the film plane, i.e., angle of view.
Projection parameters
Projection type
View frustum
View plane distance
Window or angle of view
Front and back clipping plane distances
near (>0), far (>0)
Chap 3, View Pipeline (CG-U) 10

11. Viewing Pipeline - 1 Review….. View transform
Tview
world coord. sys. -> view coord. sys.
For simpler and faster clipping and projection
Chap 3, View Pipeline (CG-U) 11

12. Viewing Pipeline - 2 Review….. View transform
View transform matrix Tview
[ xv, yv, zv, 1] = Tview [xw, yw, zw, 1]T
Tview can be written as Tview = B T ,
where
T(-C) translates the world coordinate origin to C.
B rotates any vector expressed in the world coordinate system into the view coordinate
system by mapping (u,v,n) to (e1,e2,e3).
Chap 3, View Pipeline (CG-U) 12

13. Viewing Pipeline - 3 View transform
Chap 3, View Pipeline (CG-U) 13

14. Viewing Pipeline - 4 View transform
Chap 3, View Pipeline (CG-U) 14

15. Viewing Pipeline - 5 Review….. View projection
view coord. sys. -> screen coord. sys.
Convert view volume to normalized view volume and then do orthogonal projection
Why and How? -1
Mapping view volume
to normalized view volume.
Easy view-volume clipping
and projection, and
Standard sized window. -1
Chap 3, View Pipeline (CG-U) 15

16. Projection normalization -1
Eye coordinate system:
right-handed system
Viewing direction: -z
Normalized device coord.:
left-handed system
Viewing direction: +z
Chap 3, View Pipeline (CG-U) 16

17. Projection normalization -2=
Orthogonal
projection of the distorted object
Perspective
projection
Chap 3, View Pipeline (CG-U) 17

18. Projection normalization -3
Converts all projections into orthogonal projection by
distorting the objects such that the orthogonal projection of the distorted objects is the same as the desired projection of the original objects.
Converting view volume to normalized view volume, followed by the orthogonal projection
Reasons
Easy for view volume clipping
Normalization let us clip against a simple cube regardless of type of projection
Chap 3, View Pipeline (CG-U) 18

19. Projection normalization -4
Reasons
Easy for view volume/back-face culling
The nonsingular homogeneous transformation retains meaningful depth value along the projectors that is necessary for HSR.
Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume, allowing us to use standard transformations in the pipeline
Chap 3, View Pipeline (CG-U) 19

20. Projection normalization -5
Reasons
This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping
Delay final “projection” until end
Important for hidden-surface removal to retain depth information as long as possible
Chap 3, View Pipeline (CG-U) 20

21. Projection normalization -6
Perspective projection is broken into two parts
Perspective transformation + perspective division (back to 3D)
Converts the view volume to a normalized view volume, called canonical view volume
Objects are distorted in a manner that yields the desired projection through the second step.
Orthogonal projection of the distorted objects
Need to make sure
Correct projection
Meaningful depth
Chap 3, View Pipeline (CG-U) 21

22. Projection normalization -7
Perspective
transformation =
Perspective projection
= Perspective transform +
perspective division +
orthographic projection
Orthogonal
projection of the distorted object
Chap 3, View Pipeline (CG-U) 22

23. Simple perspective projection (w/o view volume) -1
In 3D:
d is the distance
from eye to view plane
Nonlinear!!
No matrix form!! -d -d -d -d
Chap 3, View Pipeline (CG-U) 23

24. Simple perspective projection (w/o view volume) -2
Preserves lines, but it is nonlinear, not affine
No meaningful depth is preserved
Can actually decompose the perspective projection into
perspective projection (in 3D)
= perspective transform (linear, in homo. coord.)
+ perspective division + orthographic proj.
[Here, depth is still not preserved!!!]
Chap 3, View Pipeline (CG-U) 24

25. Simple perspective projection (w/o view volume) -3
Perspective transform M : in homo. coord.
Perspective division (de-homogenization)
Chap 3, View Pipeline (CG-U) 25

26. What we are going to do?
First derive the perspective transform matrix for a right view volume
Show that it does the same projection as the original perspective projection
It converts the right view volume to a normalized view volume
It preserves a meaningful depth
Extend the result to general perspective view volumes
Chap 3, View Pipeline (CG-U) 26

27. Perspective normalization transformation -1
For a right view volume specified by
Eyepoint: (0, 0, 0), view plane at z=-n
Planes: x=±z, y=±z, z=-n, z=-f (f>n) (symmetric, 45o)
View window (-n, -n,- n), (n, n, -n)
Assuming near plane distance n = view plane distance d
Consider the nonsingular matrix
α, β unspecified (but nonzero), used to transfer the near and far clipping planes to z=-1 and z=1
z=-f
(n,n,-n)
(-n,-n,-n)
z=-n
Chap 3, View Pipeline (CG-U) 27

28. Perspective normalization transformation -2
Apply N’ to point p=[x y z 1]T, we obtain p’=[x’ y’ z’ w’]T :
After dividing by w’, we have
Consider only (x”, y”),
this is the same as the simple perspective
projection!! (Note that n=d)
And z’’ has a meaningful depth!!!!
Chap 3, View Pipeline (CG-U) 28

29. Perspective normalization transformation -3
Orthographic projection of distorted objects
= Perspective projection of original objects !
If we apply an orthographic projection along z-axis to N’, we obtain a simple perspective-projection matrix.
Don’t care so is z value!!!
view plane at z=-d
Chap 3, View Pipeline (CG-U) 29

30. Perspective normalization transformation -4
Apply the above projection to p, we have
This is exactly the result
of a simple perspective
projection for view plane
at z=-n
So,
Perspective projection = perspective transformation N’ +
orthographic projection Morth
Chap 3, View Pipeline (CG-U) 30

31. Perspective normalization transformation -5
N’ is nonsingular and does the following
x= ±z is transformed to planes x”= ±n (by putting x= ±z in x”=-nx/z)
y= ±z is transformed to planes y”= ±n (by putting y= ±z in y”=-ny/z)
z=-n and z=-f are transformed by N’ to the planes that are perpendicular to z-axis, and
Chap 3, View Pipeline (CG-U) 31

32. Perspective normalization transformation -6
If we select
the transformed clipping planes become, respectively,
This choice
Preserves the depth order
Normalizes the z-values within
near and far planes to [-1,1]
Chap 3, View Pipeline (CG-U) 32

33. Perspective normalization transformation -7
Chap 3, View Pipeline (CG-U) 33

34. Perspective normalization transformation -8+
Chap 3, View Pipeline (CG-U) 34

35. Perspective normalization transformation -9+ +
Chap 3, View Pipeline (CG-U) 35

36. Perspective normalization transformation -10
The mapping is nonlinear but preserves the ordering of depths, i.e., if z1 > z2 in the original view volume, then z1”< z2” :
Chap 3, View Pipeline (CG-U) 36

37. Perspective normalization transformation -11
Eye coordinate system:
right-handed system
Viewing direction: -z
Normalized device coord.:
left-handed system
Viewing direction: +z
See OpenGL projection matrix at
Chap 3, View Pipeline (CG-U) 37

38. Perspective normalization transformation -12
z= -f z2 +z z1
z= 1
z= -n
z2’’
z1’’ +x +x
z= -1 +z -z
Z1 > Z2
Z1’’ < Z2’’

39. Perspective normalization transformation -13
Hidden surface removal works if we first apply the normalization transformation
However, the formula implies that the distances are distorted by the normalization which can cause numerical problems especially if the near distance is small
Chap 3, View Pipeline (CG-U) 39

40. Perspective normalization transformation -14
Chap 3, View Pipeline (CG-U) 40

41. Perspective normalization transformation -15
Properties of perspective transformation N
Preserves lines and flatness.
Preserves in-between-ness
If a point inside an object, the transformed point will be inside the transformed object.
Preserves depth order with its pseudo-depth.
Warps objects such that the orthographic projection of the warped object = perspective projection of the original one.
Warps the view volume to the canonical view volume
Chap 3, View Pipeline (CG-U) 41

42. Perspective normalization transformation -16
N transforms the right view volume to the canonical view volume (after perspective division), and an orthographic projection in the transformed volume yields the same image as does the perspective projection (Consider only x, y-value).
N is called perspective transformation matrix. -z +z
Chap 3, View Pipeline (CG-U) 42

43. Perspective normalization transformation -17
For an asymmetric, not right view volume (whose face planes are not x=±z, y=±z), we need to do
a shear transformation H and
a scaling S
before applying N.
Chap 3, View Pipeline (CG-U) 43

44. Perspective normalization transformation -18
The shear transformation H shears the point to by
Since z<0, (l+r)/(2n)
must be < 0 when
(l+r)<0 to make
a move to the right.
(l+r)/(2n) must be >0
when (l+r)>0 to make
a left move.
Chap 3, View Pipeline (CG-U) 44

45. Perspective normalization transformation -19
Review of shear transformation
shear the point to
Chap 3, View Pipeline (CG-U) 45

46. Perspective normalization transformation -20
Chap 3, View Pipeline (CG-U) 46

47. Perspective normalization transformation -21
The scaling transformation S
Scale the sheared view volume defined by
to a symmetric-right view volume
without changing the near and far planes.
Scaling matrix (derived using 4 corners of the window on the near plane)
Positive scaling factors!!
Chap 3, View Pipeline (CG-U) 47

48. Perspective normalization transformation -22
Scale factor:
divide by (r-l)/2,
then multiply by n
Chap 3, View Pipeline (CG-U) 48

49. Perspective normalization transformation -23
Projection matrix is
Precisely the projection matrix that OpenGL creates when gluFrustum() is executed.
Chap 3, View Pipeline (CG-U) 49

50. Perspective normalization transformation -24
Chap 3, View Pipeline (CG-U) 50

51. Another derivation of N -1 from pseudo-depth
Simple perspective transformation
A pseudo-depth is desired
Has a meaningful depth function with the same denominator as x and y, and hence apply the same transformation matrix. so try
No depth !!
Chap 3, View Pipeline (CG-U) 51

52. Another derivation of N -2 from pseudo-depth
For the transformation
We choose and such that the pseudo-depth varies between -1 and 1.
Depth=-1 for z=-n, Depth=1 for z=-f
Chap 3, View Pipeline (CG-U) 52

53. OpenGL Canonical view volume
A cube whose center is at the origin of the window coordinate and whose faces are given by 6 planes: x=1, -1, y=1, -1, z=1, -1.
Canonical view volume is the default in OpenGL (result of glFrustum() and glOrtho()), or obtained by
glMatrixMode(GL_projection);
glLoadIdentity();
glOrtho(-1.0, 1.0, -1.0, 1.0, -1.0, 1.0);
z=-1.0 is the near plane
z= 1.0 is the far plane
Chap 3, View Pipeline (CG-U) 53

54. OpenGL Perspective projection -1
For general symmetric view volume
gluPerspective(fovy, aspect, near, far)
Produces a projection matrix that converts the symmetric view volume to the symmetric-right one by a scaling transformation S, and to a canonical view volume by a N.
Produces P = NS
Chap 3, View Pipeline (CG-U) 54

55. OpenGL Perspective projection -2
view plane
aspect = w/h
Chap 3, View Pipeline (CG-U) 55

56. OpenGL Perspective projection -3
For asymmetric volume
glFrustum(l, r, b, t, n, f)
Produces a projection matrix that converts the asymmetric view volume to the symmetric-right one by a shear transformation H followed by a scaling transformation S, and to a canonical view volume by a N
Produces P = NSH
Chap 3, View Pipeline (CG-U) 56

57. Orthogonal projections -1
What is the projection matrix?
Move the center of the specified view volume to the center of the canonical view volume by
Scale the sides by
Chap 3, View Pipeline (CG-U) 57

58. Orthogonal projections -2
Projection matrix
Chap 3, View Pipeline (CG-U) 58

59. OpenGL Orthogonal projections -1
OpenGL provides only orthographic projection
glOrtho(xmin, xmax, ymin, ymax, near, far)
Parameters are identical to those of glFrustum()
zmax
zmax
Zmax = -far
zmin = -near
zmin
zmin
Chap 3, View Pipeline (CG-U) 59

60. OpenGL Orthogonal projections -2
Converting general view volume defined by
glOrtho(xmin, xmax, ymin, ymax, near, far);
Produces a projection matrix that transforms the view volume to the canonical view volume.
Vertices inside (outside) the original view volume are transformed by the projection matrix to the vertices inside (outside) the canonical view volume.
,-1)
(-1,-1,
Chap 3, View Pipeline (CG-U) 60

61. OpenGL Projection normalization
Clip coordinates (homogeneous): has depth
Normalized device coord. (3D): has depth
Window coordinates (2D): has no depth
Projection in OpenGL
Part 1: results in a projection matrix.
Followed a perspective division to convert vertices to normalized device coordinates.
Part 2: do nothing, neglect z.
Converts to window coordinates.
Chap 3, View Pipeline (CG-U) 61

62. Projection normalization Oblique projection
Not available in OpenGL.
Characterized by the angle between the projector and the view plane. Assuming that
Near and far planes parallel to the view plane
Other faces parallel to the projector’s direction.
Chap 3, View Pipeline (CG-U) 62

63. Oblique projection Projection matrix -1
Consider the top and side view
Shearing
Chap 3, View Pipeline (CG-U) 63

64. Oblique projection Projection matrix -2
Shearing
Chap 3, View Pipeline (CG-U) 64

65. Oblique projection Projection matrix -3
For view volume with unit length and origin as center
P can be broken into two matrices
Morth is orthographic projection, H() is a shear matrix
Chap 3, View Pipeline (CG-U) 65

66. Oblique projection Projection matrix -4
Morth
Chap 3, View Pipeline (CG-U) 66

67. Oblique projection Projection matrix -5
For general oblique projection
A shear of the object by H(θ,Φ)
A translation and a scaling (convert the sheared view volume to the canonical view volume)
An orthographic projection
Projection matrix
Chap 3, View Pipeline (CG-U) 67

68. View volume clipping -1 Clip against canonical view volume?
It is parameter free, uses only -1 and 1.
Planes are aligned with coordinate axes.
Why in homogeneous space?
Clipping in HS isn’t completely necessary, but it makes the clipping clean, fast, and simple, at almost no cost.
Done after perspective division
Signs of x (or y, z) and w are lost. Can tell only if x and w have the same or opposite signs.
Chap 3, View Pipeline (CG-U) 68

69. View volume clipping -2
Situations that necessitate clipping in HS involve
Particular transformation of objects or the construction of rational surfaces such that the point has a negative w, even the point is in front of the eye.
Clipping in 3D is correct only when w>0
In HS: two clip regions A (w>0) & B ( w<0)
Negate points with w<0 and clip all against A.
Chap 3, View Pipeline (CG-U) 69

70. View volume clipping -2 Uses Cyrus-Beck clipper (a topic in Chap. 4)
Chap 3, View Pipeline (CG-U) 70

71. Viewing Pipeline Summary
projection
View transform
View-Volume
Clipping in
homogeneous
space
Window-viewport
mapping
Chap 3, View Pipeline (CG-U) 71