1. Viewing III Week 4, Wed Jan 27

2. News extra TA office hours in lab 005
Tue 2-5 (Kai)
Wed 2-5 (Garrett)
Thu 1-3 (Garrett), Thu 3-5 (Kai)
Fri 2-4 (Garrett)
Tamara's usual office hours in lab
Fri 4-5

3. Review: Convenient Camera Motion
rotate/translate/scale versus
eye point, gaze/lookat direction, up vector y
lookat x
Pref
WCS
view up z
eye
Peye

4. Review: World to View Coordinates
translate eye to origin
rotate view vector (lookat – eye) to w axis
rotate around w to bring up into vw-plane y z x
WCS v u
VCS
Peye w
Pref up
view
eye
lookat

5. Review: W2V vs. V2W MW2V=TR we derived position of camera in world
invert for world with respect to camera
MV2W=(MW2V)-1=R-1T-1

6. Review: Graphics Cameras
real pinhole camera: image inverted
eye
point
image
plane
computer graphics camera: convenient equivalent
stand at p and look through hole. anything in cone visible
shrink hole to a point, cone collapses to line (ray)
eye
point
center of
projection
image
plane

7. Review: Projective Transformations
planar geometric projections
planar: onto a plane
geometric: using straight lines
projections: 3D -> 2D
aka projective mappings
counterexamples?
nonplanar: planetarium, omnimax
nonstraight lines: Mercator projection of maps, stereographic projection
non3D: hypercube from 4D

8. Projective Transformations
properties
lines mapped to lines and triangles to triangles
parallel lines do NOT remain parallel
e.g. rails vanishing at infinity
affine combinations are NOT preserved
e.g. center of a line does not map to center of projected line (perspective foreshortening)

9. Perspective Projection
project all geometry
through common center of projection (eye point)
onto an image plane x -z y z z x x

10. Perspective Projection
projection plane
center of projection
(eye point)
how tall should this bunny be?

11. Basic Perspective Projection
similar triangles y
P(x,y,z)
P(x’,y’,z’) z
z’=d
but
nonuniform foreshortening
not affine

12. Perspective Projection
desired result for a point [x, y, z, 1]T projected onto the view plane:
what could a matrix look like to do this?

13. Simple Perspective Projection Matrix

14. Simple Perspective Projection Matrix
is homogenized version of
where w = z/d

15. Simple Perspective Projection Matrix
is homogenized version of
where w = z/d

16. Perspective Projection
expressible with 4x4 homogeneous matrix
use previously untouched bottom row
perspective projection is irreversible
many 3D points can be mapped to same (x, y, d) on the projection plane
no way to retrieve the unique z values

17. Moving COP to Infinity as COP moves away, lines approach parallel
when COP at infinity, orthographic view

18. Orthographic Camera Projection
camera’s back plane parallel to lens
infinite focal length
no perspective convergence
just throw away z values

19. Perspective to Orthographic
transformation of space
center of projection moves to infinity
view volume transformed
from frustum (truncated pyramid) to parallelepiped (box) x x
Frustum
Parallelepiped -z -z

20. perspective view volume orthographic view volume
View Volumes
specifies field-of-view, used for clipping
restricts domain of z stored for visibility test
perspective view volume
orthographic view volume
x=left
x=right
y=top
y=bottom
z=-near
z=-far x
VCS z y z

21. Canonical View Volumes
standardized viewing volume representation
perspective orthographic
orthogonal
parallel
x or y = +/- z
x or y
x or y
back plane
back plane 1
front plane
front plane -z -z -1 -1

22. Why Canonical View Volumes?
permits standardization
clipping
easier to determine if an arbitrary point is enclosed in volume with canonical view volume vs. clipping to six arbitrary planes
rendering
projection and rasterization algorithms can be reused

23. Normalized Device Coordinates
convention
viewing frustum mapped to specific parallelepiped
Normalized Device Coordinates (NDC)
same as clipping coords
only objects inside the parallelepiped get rendered
which parallelepiped?
depends on rendering system

24. Normalized Device Coordinates
left/right x =+/- 1, top/bottom y =+/- 1, near/far z =+/- 1
Camera coordinates
NDC x x
x=1
right
Frustum -z z
left
x= -1
z= -1
z=1
z=-n
z=-f

25. Understanding Z z axis flip changes coord system handedness
RHS before projection (eye/view coords)
LHS after projection (clip, norm device coords)
VCS
NDCS
y=top y
(1,1,1)
x=left y z z
(-1,-1,-1)
x=right x x
z=-far
y=bottom
z=-near

26. perspective view volume
Understanding Z
near, far always positive in OpenGL calls
glOrtho(left,right,bot,top,near,far);
glFrustum(left,right,bot,top,near,far);
glPerspective(fovy,aspect,near,far);
perspective view volume
x=left
x=right
y=top
y=bottom
z=-near
z=-far x
VCS y
orthographic view volume
y=top
x=left y z
x=right
VCS x
z=-far
y=bottom
z=-near

27. Understanding Z why near and far plane? near plane: far plane:
avoid singularity (division by zero, or very small numbers)
far plane:
store depth in fixed-point representation (integer), thus have to have fixed range of values (0…1)
avoid/reduce numerical precision artifacts for distant objects

28. Orthographic Derivation
scale, translate, reflect for new coord sys
VCS x z
NDCS y
(-1,-1,-1)
(1,1,1)
y=top
x=left y z
x=right x
z=-far
y=bottom
z=-near

29. Orthographic Derivation
scale, translate, reflect for new coord sys
VCS x z
NDCS y
(-1,-1,-1)
(1,1,1)
y=top
x=left y z
x=right x
z=-far
y=bottom
z=-near

30. Orthographic Derivation
scale, translate, reflect for new coord sys

31. Orthographic Derivation
scale, translate, reflect for new coord sys
VCS
y=top
x=left y z
x=right x
z=-far
y=bottom
z=-near
same idea for right/left, far/near

32. Orthographic Derivation
scale, translate, reflect for new coord sys

33. Orthographic Derivation
scale, translate, reflect for new coord sys

34. Orthographic Derivation
scale, translate, reflect for new coord sys

35. Orthographic Derivation
scale, translate, reflect for new coord sys

36. Orthographic OpenGL glMatrixMode(GL_PROJECTION); glLoadIdentity();
glOrtho(left,right,bot,top,near,far);

37. Demo Brown applets: viewing techniques parallel/orthographic cameras
projection cameras

38. Projections II

39. Asymmetric Frusta our formulation allows asymmetry why bother? x x
right
right
Frustum
Frustum
Q: why would we want to do this?
A: stereo viewing.
can’t just rotate because axes of projection differ
then z-order could differ, orientation of shapes differ.
so use skewed frustum. -z -z
left
left
z=-n
z=-f

40. Asymmetric Frusta our formulation allows asymmetry
why bother? binocular stereo
view vector not perpendicular to view plane
Left Eye
Q: why would we want to do this?
A: stereo viewing.
can’t just rotate because axes of projection differ
then z-order could differ, orientation of shapes differ.
so use skewed frustum.
Right Eye

41. Simpler Formulation left, right, bottom, top, near, far
nonintuitive
often overkill
look through window center
symmetric frustum
constraints
left = -right, bottom = -top

42. Field-of-View Formulation
FOV in one direction + aspect ratio (w/h)
determines FOV in other direction
also set near, far (reasonably intuitive) x w
fovx/2 h
Frustum -z 
fovy/2
z=-n
z=-f

43. Perspective OpenGL glMatrixMode(GL_PROJECTION); glLoadIdentity();
glFrustum(left,right,bot,top,near,far); or
glPerspective(fovy,aspect,near,far);

44. Demo: Frustum vs. FOV http://www.xmission.com/~nate/tutors.html
Nate Robins tutorial (take 2):

45. Projective Rendering Pipeline
object
world
viewing
O2W
W2V
V2C
OCS
WCS
VCS
projection
transformation
modeling
transformation
viewing
transformation
clipping
C2N
CCS
OCS - object/model coordinate system
WCS - world coordinate system
VCS - viewing/camera/eye coordinate system
CCS - clipping coordinate system
NDCS - normalized device coordinate system
DCS - device/display/screen coordinate system
perspective divide
normalized
device
N2D
NDCS
viewport
transformation
device
DCS

46. Projection Warp warp perspective view volume to orthogonal view volume
render all scenes with orthographic projection!
aka perspective warp x x
z=d
z=d
z=
z=0

47. Perspective Warp perspective viewing frustum transformed to cube
orthographic rendering of cube produces same image as perspective rendering of original frustum

48. Predistortion

49. Projective Rendering Pipeline
object
world
viewing
O2W
W2V
V2C
OCS
WCS
VCS
projection
transformation
modeling
transformation
viewing
transformation
clipping
C2N
CCS
OCS - object/model coordinate system
WCS - world coordinate system
VCS - viewing/camera/eye coordinate system
CCS - clipping coordinate system
NDCS - normalized device coordinate system
DCS - device/display/screen coordinate system
perspective divide
normalized
device
N2D
NDCS
viewport
transformation
device
DCS

50. Separate Warp From Homogenization
viewing
clipping
normalized
device
VCS
V2C
CCS
C2N
NDCS
projection
transformation
perspective
division
alter w
/ w
warp requires only standard matrix multiply
distort such that orthographic projection of distorted objects is desired persp projection
w is changed
clip after warp, before divide
division by w: homogenization

51. Perspective Divide Example
specific example
assume image plane at z = -1
a point [x,y,z,1]T projects to [-x/z,-y/z,-z/z,1]T 
[x,y,z,-z]T -z

52. Perspective Divide Example
after homogenizing, once again w=1
projection
transformation
perspective
division
alter w
/ w

53. Perspective Normalization
matrix formulation
warp and homogenization both preserve relative depth (z coordinate)

54. Demo Brown applets: viewing techniques parallel/orthographic cameras
projection cameras