1. Viewing Transformations II
CS 455 – Computer Graphics
Viewing Transformations II

2. Arbitrary Views
Suppose we want to look at the environment from somewhere other than along the z-axis
How do we do this?
What is the perspective matrix?
Where should an object project onto our new viewplane?
Solution:
set up our arbitrary view
transform it into the situation we know how to handle (i.e., viewpoint on the +z axis looking at the origin)
apply that transformation to the environment
then apply the perspective computation

3. Specifying and Arbitrary View
What do we need to specify and arbitrary view?
1. Projection plane (or View plane)
Specified with a point and a normal
We will call the point on the plane the “View Reference Point” or VRP
We will call the normal to the plane the “View Plane Normal” or VPN
We can define these anywhere in world space
Viewplane
VRP
VPN

4. Specifying and Arbitrary View
What else?
2. A window in the infinite viewplane
We want to specify a subset of the viewplane that we will project onto the screen
Objects outside the window will not be seen
How do we define that window?
Very difficult to do in world coordinates!
Viewplane
Window
VRP
VPN

5. Specifying the Window
We would like to specify the window relative to the VRP (some number of units above, below, left and right)
To do this, we must create a new coordinate system
This will be called the “View-Reference Coordinate System” (VRC)
The origin of the VRC is the VRP
One axis is the VPN
A second axis will be the “up” axis
This will need to be specified by the programmer
It is simply the direction that is to be “up”
i.e. the up direction of the eye or camera

6. Specifying the Window
The “view up” vector (VUP) is projected onto the viewplane to get the second (V) axis
This allows the programmer to specify any vector - not just a vector exactly perpendicular to N
VUP projected onto Viewplane
(V axis)
Viewplane
VUP
VRP (VRC origin)
VPN (N axis)

7. Specifying the Window
The third axis (U) is obtained by taking the cross product of V and N
VUP projected onto Viewplane
(V axis)
Viewplane
U axis
VRP (VRC origin)
VPN (N axis)

8. Computing the VRC system
The actual way this is done is:
VRP and VPN are specified
N = VPN
View Up is specified
U = VUP X N
V = N X U V
Viewplane
VUP U N

9. Specifying the Window
With the VRC system defined, we can now specify the window
Window = (umin, vmin) -> (umax, vmax)
the Center of the Window (CW) is midway between
(umin, vmin) and (umax, vmax) V
(umax, vmax)
Viewplane CW U
(umin, vmin) N

10. Specifying an Arbitrary View
Back to specifying our arbitrary view
What else do we need?
3. Projection type
Parallel
Perspective V
(umax, vmax)
Viewplane CW U
(umin, vmin) N

11. Specifying an Arbitrary View
What else do we need?
4. Projection Reference Point (PRP)
Specified in VRC
For perspective projection, it is the center of projection
Defines the view volume
as a semi-infinite
pyramid whose apex
is the PRP V
Viewplane U N
PRP - Center of Projection

12. Specifying an Arbitrary View
For parallel projection, the direction of projection is obtained from the PRP and the CW
The view volume is the infinite parallelepiped whose sides are parallel to the direction of projection (DOP) and pass through the borders of the window
Viewplane V
DOP CW U N
PRP

13. Specifying an Arbitrary View
And Finally:
5. Front and back clipping planes
If desired
Limits the view volume
to a finite volume and
clips objects that don’t
fall within it
Perspective Viewing
Back Clipping Plane
Window
Front
Clipping
Plane
VRP B F

14. Specifying an Arbitrary View
Parallel Viewing
Back Clipping Plane
Window
Front
Clipping
Plane
VRP B F

15. Specifying an Arbitrary View Summary
Need to specify:
View Reference Point - VRP
Projection Reference Point - PRP
View Plane Normal - VPN
View Up vector - VUP
Window bounds
Can be specified either as
(umin, vmin), (umax, vmax) or
FieldOfViewx, FieldOfViewy
Projection Type

16. Specifying an Arbitrary View Summary
Generally this is simplified for the user to:
LookAt Point - VRP
LookFrom Point - PRP, view point, or center of projection
View Up vector - VUP
Field of View - gives the window size (could be different for vertical and horizontal, or could be the same)
Projection Type
Then with these quantities specified:
VRP = LookAt
PRP = LookFrom
VPN = LookFrom - LookAt
N = VPN
U = VUP X N
V = N X U

17. Arbitrary View So, we have: Back plane V Viewplane VUP
Field of View in U
VUP
LookAt U
Front plane
Field of View in V
LookFrom N

18. Arbitrary View
In world space we have: y v u n x z

19. Transforming the Arbitrary View
Now we need to put the arbitrary view into a situation we know how to handle,
i.e., LookFrom on the +z axis, LookAt at the origin
How?
Perform a sequence of transformations that will perform this

20. Transforming the Arbitrary View
Step 1: Translate LookAt to origin
T(-xvrp, -yvrp, -zvrp) y v y v’ n u n’ x u’ x z z

21. Transforming the Arbitrary View
Step 2: Rotate about y to bring n into the yz plane
Ry(Q) y y v’
v’’ n’
u’’ u’ x
n’’ x z z

22. Transforming the Arbitrary View
What is Q?
Looking down the y axis: x Q
n’z n’ z

23. Transforming the Arbitrary View
Step 3: Rotate about x to bring n coincident with the z axis
Rx(F) y y
v’’
v’’’
u’’
n’’ x
n’’’
u’’’ x z z

24. Transforming the Arbitrary View
What is F?
Looking down the x axis: y
n’’z z F
n’’

25. Transforming the Arbitrary View
Step 4: Rotate about z to bring v coincident with the y axis and u coincident with the x axis
Rz(g) y y
v’’’
viv
n’’’
niv
u’’’
uiv x x z z

26. Transforming the Arbitrary View
What is g?
Looking down the z axis: y
u’’’x x g
u’’’

27. Transforming the Arbitrary View
Now we have the arbitrary view transformed into a situation we know how to handle
At this point, we can apply the perspective transformation, then render
Final transformation: y
viv
niv
uiv x z

28. Applying the Arbitrary View
Once we have Marb_view computed, we multiply each point in each object by that Marb_view
This has the effect of transforming our objects around so that the image we render is the image we would see from the given LookFrom point
Marb_view is complex to compute - is there a better way?
Yes! Remember linear algebra…..
Transforming from one space to another

29. Transforming between Spaces
Given our 3D cartesian coordinate system (x, y, z) and a separate 3D system defined somewhere in world space, how do we transform between them? y v u n x z

30. Transforming between Spaces
Given
u = (u1, u2, u3)
v = (v1, v2, v3)
n = (n1, n2, n3)
And we want
u to line up with the x axis
v to line up with the y axis
n to line up with the z axis z y x v u n

31. Transforming between Spaces
Step 1: Translate the VRP to the World Coordinate origin
Step 2: Apply the following matrix to transform from uvn space to xyz space y v u n x z y v u n x z

32. Canonical View Volumes
In order to simplify clipping against a view volume, the notion of a canonical view volume is used.
It is just a particular view volume that allows for clipping to be done very quickly
We will take our viewing situation, and go through a sequence of transformations to place us in the canonical view volume.
Objects are then clipped against this view volume, then rendered

33. Canonical View Volumes
Parallel canonical view volume
Defining Planes:
x = -1
x = 1
y = -1
y = 1
z = 0
z = -1 y
Back plane x z
Front plane

34. Canonical View Volumes
Perspective canonical view volume
Defining Planes:
x = -z
x = z
y = -z
y = z
z = -zmin
z = -1
Back plane y
Front plane x z

35. Canonical View Volumes
Perspective Canonical view volume
Looking down x: Looking down y: y -x -1 -1 -z -z x -y
zmin
zmin

36. Transforming to Canonical View Volume
How do we go from our situation (eye on +z looking at origin) to the canonical view volume? y y ?
viv x
niv z
uiv x
LookFromiv
More transformations! z

37. Transforming to Canonical View Volume
Step 1: translate Projection Reference Point (LookFrom) to origin z y x
uiv
viv
niv z y x
uiv
viv
niv
vrpzv
prpiv
T(0, 0, -prpzIV)

38. Transforming to Canonical View Volume
Step 2: Scale so that the View Volume becomes the Canonical View Volume
Part a: scale in x and y such that the view plane has dimensions 2*vrpz
current viewplane dimensions: (umax - umin) by (vmax - vmin)

39. Canonical View Volumes
Step 2a
Looking down x:
Before 1st scale After 1st scale y -z -y
vrpz
vrpz+B
vrpz+F y -z -y
vrpz
vrpz+B
vrpz+F

40. Transforming to Canonical View Volume
Step 2b: Scale so that the View Volume sides have correct slope, and so that the z coordinate of the back clipping plane is -1
Back clipping plane is located at vrpz + B, so the scale factor is

41. Transforming to Canonical View Volume
So the total scale is,

42. Canonical View Volumes
Step 2a
Looking down x:
Before 2nd scale After 2nd scale y
vrpz y -z -y
viewplane
z=-1
z=zmin -z
vrpz+B -y
vrpz+F

43. Canonical View Wrap Up Now we apply the perspective projection
The final matrix sequence is:
or (for change of basis): y x z

44. Clipping against the Canonical View
Once we are in the canonical view, clipping is much easier.
Extend the 2D Cohen-Sutherland clipping algorithm to 3D
Use a 6-bit code
Bit codes signify:
bit 1 - point above view volume (y > -z)
bit 2 - point below view volume (y < z)
bit 3 - bit right of view volume (x > -z)
bit 4 - bit left of view volume (x < z)
bit 5 - bit behind view volume (z < -1)
bit 6 - bit in front of view volume (z > zmin)

45. Example Given: Where does (-1, 1, 0) project? LookFrom at (2, 0, 2)
LookAt at (1, 0, 1)
ViewUp (0, 1, 0)
ViewAngle 45 degrees
Viewport (500, 500)
Where does (-1, 1, 0) project? z y x v u n
(-1, 1, 0)

46. Example z y x v u n
(-1, 1, 0)
First, compute what we need: N: U: V:

47. Example Next, compute transformation matrices: Translate:z y x v u n
(-1, 1, 0)
Next, compute transformation matrices:
Translate:
Change of Basis:

48. Example z y x v u n
(-1, 1, 0)
Multiply matrices together:

49. Example The UVN coordinate system will be lined up with XYZ.
(-1, 1, 0)
The UVN coordinate system will be
lined up with XYZ.
Where will (-1, 1, 0) now be?

50. Example Where does the LookFrom point transform to? y v’ x n’ u’ z
(-.707, 1, ) x n’ u’ z

51. Example Next, compute the Perspective Matrix: What is d? y v’ x n’ u’
(-.707, 1, ) x n’ u’
What is d? z

52. Example Where does the point project onto the viewplane using they
Where does the point project
onto the viewplane using the
perspective projection? v’
(-.283, 0.4, 0) x n’ u’ z

53. Example Next: Window-to-viewport What are the window bounds? y v’
(-.283, 0.4, 0)
Looking down y x x
Q = 22.5o z
xwin n’ u’ z
So the window is defined between (-0.59, 0.59)
Note: the projected point falls in that bound, so it will appear in the viewport

54. Example Window-to-viewport First, translate:y
Window-to-viewport
First, translate: v’
(.337, 1.02, 0) x n’ u’ z
applying to the point gives:

55. Example y
Window-to-viewport
Next, scale: v’
(.337, 1.02, 0) x n’ u’ z

56. Example Window-to-viewport Applying to the point:
(.307, .99, 0) x n’ u’ z
So, the point projects to
(124, 399) in the viewport

57. Quiz
What does the perspective matrix look like for the situation where the LookFrom point is on the -x axis, and the LookAt point is the origin?