1. Viewing 3 http://www.ugrad.cs.ubc.ca/~cs314/Vjan2016
University of British Columbia
CPSC 314 Computer Graphics
Jan-Apr 2016
Viewing 3

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. perspective view volume
Understanding Z
near, far always positive in GL calls
THREE.OrthographicCamera(left,right,bot,top,near,far);
mat4.frustum(left,right,bot,top,near,far, projectionMatrix);
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

9. 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

10. 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

11. 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

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

13. 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

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

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

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

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

18. 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

19. 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

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

21. Predistortion

22. 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

23. 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

24. 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

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

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

27. Perspective To NDCS Derivation
VCS
y=top
NDCS
x=left y
(1,1,1) y z
(-1,-1,-1) x z
z=-near
y=bottom
z=-far x
x=right

28. Perspective Derivation
simple example earlier:
complete: shear, scale, projection-normalization

29. Perspective Derivation
earlier:
complete: shear, scale, projection-normalization

30. Perspective Derivation
earlier:
complete: shear, scale, projection-normalization

31. Perspective Derivation

32. Perspective Derivation
similarly for other 5 planes
6 planes, 6 unknowns

33. 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

34. NDC to Device Transformation
map from NDC to pixel coordinates on display
NDC range is x = , y = , z =
typical display range: x = , y =
maximum is size of actual screen
z range max and default is (0, 1), use later for visibility
gl.viewport(0,0,w,h);
gl.depthRange(0,1); // depth = 1 by default x
500 y y -1 x
viewport
NDC 1 -1 1
300

35. Origin Location yet more (possibly confusing) conventions
GL origin: lower left
most window systems origin: upper left
then must reflect in y
when interpreting mouse position, have to flip your y coordinates x
500 y y -1 x
viewport
NDC 1 -1 1
300

36. N2D Transformation general formulation
reflect in y for upper vs. lower left origin
scale by width, height, depth
translate by width/2, height/2, depth/2
FCG includes additional translation for pixel centers at (.5, .5) instead of (0,0) x y
viewport
NDC
500
300 -1 1
height
width

37. N2D Transformation Display z range is 0 to 1.
gl.depthRange(n,f) can constrain further, but depth = 1 is both max and default
reminder:
NDC z range is -1 to 1 x y
viewport
NDC
500
300 -1 1
height
width

38. Device vs. Screen Coordinates
viewport/window location wrt actual display not available within GL
usually don’t care
use relative information when handling mouse events, not absolute coordinates
could get actual display height/width, window offsets from OS
loose use of terms: device, display, window, screen... x
1024 x
500 y
y offset y
display
height
x offset
viewport
viewport
300
display
768
display width

39. Projective Rendering Pipeline
glVertex3f(x,y,z)
object
world
viewing
O2W
W2V
V2C
alter w
OCS
WCS
VCS
glFrustum(...)
projection
transformation
modeling
transformation
viewing
transformation
clipping
glTranslatef(x,y,z)
glRotatef(a,x,y,z)
....
gluLookAt(...)
C2N
/ w
CCS
perspective
division
normalized
device
OCS - object coordinate system
WCS - world coordinate system
VCS - viewing coordinate system
CCS - clipping coordinate system
NDCS - normalized device coordinate system
DCS - device coordinate system
glutInitWindowSize(w,h)
glViewport(x,y,a,b)
N2D
NDCS
viewport
transformation
device
DCS

40. Coordinate Systems viewing (4-space, W=1) clipping
(4-space parallelepiped, with COP moved backwards to infinity
projection matrix
normalized device
(3-space parallelepiped)
divide by w
device
(3-space parallelipiped)
scale &
translate
framebuffer

41. Perspective Example view volume left = -1, right = 1 tracks in VCS:
bot = -1, top = 1
near = 1, far = 4
tracks in VCS:
left x=-1, y=-1
right x=1, y=-1
x=-1
x=1 1
ymax-1
z=-4
real midpoint
z=-1 -1 -1 1 -1
xmax-1 x
NDCS
(z not shown)
DCS
(z not shown) z
VCS top view

42. Perspective Example view volume left = -1, right = 1 bot = -1, top = 1
near = 1, far = 4

43. Perspective Example
/ w