http://www.ugrad.cs.ubc.ca/~cs314/Vjan2005

1
Projections IIWeek 4, Wed Jan 26

• http//www.ugrad.cs.ubc.ca/cs314/Vjan2005

2
Reading (Mon and today)

• FCG
• Section 5.3.1
• rest of Chapter 6
• RB
• rest of Chapter Viewing
• rest of Appendix Homogeneous Coords

3
Review Graphics Cameras

• real pinhole camera image inverted

eye point
image plane

• computer graphics camera convenient equivalent

eye point
center of projection
image plane
4
Review Basic Perspective Projection
similar triangles
P(x,y,z)
y
P(x,y,z)
z
zd
homogeneous coords
5
Review Orthographic Cameras

• center of projection at infinity
• no perspective convergence
• just throw away z values

6
Review Transforming View Volumes
NDCS
y
(1,1,1)
z
(-1,-1,-1)
x
7
Review Ortho to NDC Derivation

• scale, translate, reflect for new coord sys

VCS
ytop
xleft
y
z
xright
x
z-far
ybottom
z-near
8
NDC to Viewport Transformation

• generate pixel coordinates
• map x, y from range 11 (NDC) to pixel
  coordinates on the display
• involves 2D scaling and translation

y
display
x
viewport
9
NDC to Viewport Transformation

• 2D scaling and translation

(1,1)
(w,h)
DCS
b
NDCS
a
y

• (-1,-1)

x
(0,0)
OpenGL
glViewport(x,y,a,b)
default
glViewport(0,0,w,h)
10
Origin Location

• yet more possibly confusing conventions
• OpenGL lower left
• most window systems upper left
• often have to flip your y coordinates
• when interpreting mouse position

11
Perspective Example
view volume left -1, right 1 bot -1,
top 1 near 1, far 4
tracks in VCS left x-1, y-1 right
x1, y-1
x1
x-1
1
ymax-1
z-4
realmidpoint
-1
z-1
1
-1
xmax-1
0
-1
0
x
NDCS (z not shown)
DCS (z not shown)
z
VCStop view
12
Viewing Transformation
y
image plane
VCS
z
OCS
z
y
Peye
y
x
x
WCS
object
world
viewing
VCS
OCS
WCS
OpenGL ModelView matrix
13
Projective Rendering Pipeline
object
world
viewing
alter w
WCS
VCS
OCS
projection transformation
clipping
/ 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

NDCS
device
DCS
14
Projective Rendering Pipeline
object
world
viewing
alter w
WCS
VCS
OCS
projection transformation
clipping
/ 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

NDCS
device
DCS
15
Perspective Projection

• specific example
• assume image plane at z -1
• a point x,y,z,1T projects to -x/z,-y/z,-z/z,1T
  ?
• x,y,z,-zT

-z
16
Perspective Projection
projection transformation
perspective division
alter w
/ w
17
Canonical View Volumes

• standardized viewing volume representation
• orthographic
  perspective
• orthogonal
• parallel

x or y /- z
x or y
x or y
backplane
backplane
1
frontplane
FrontPlane
-z
-z
-1
-1
18
Why Canonical View Volumes?

• permits standardization
• clipping
• easier to determine if an arbitrary point is
  enclosed in volume
• consider clipping to six arbitrary planes of a
  viewing volume versus canonical view volume
• rendering
• projection and rasterization algorithms can be
  reused

19
Projection Normalization

• one additional step of standardization
• warp perspective view volume to orthogonal view
  volume
• render all scenes with orthographic projection!

x
x
zd
zd
z0
z?
20
Predistortion
21
Perspective Normalization

• perspective viewing frustum transformed to cube
• orthographic rendering of cube produces same
  image as perspective rendering of original frustum

22
Demos

• Tuebingen applets from Frank Hanisch
• http//www.gris.uni-tuebingen.de/projects/grdev/do
  c/html/etc/AppletIndex.htmlTransformationen

23
Perspective Warp

• matrix formulation
• preserves relative depth (third coordinate)
• what does mean?

24
Projection Normalization
normalized device
clipping
viewing
CCS
VCS
NDCS
projection transformation
perspective division
alter w
/ w

• distort such that orthographic projection of
  distorted objects is desired persp projection
• separate division from standard matrix multiplies
• clip after warp, before divide
• division normalization

25
Projective Rendering Pipeline
glVertex3f(x,y,z)
object
world
viewing
alter w
WCS
VCS
OCS
glFrustum(...)
projection transformation
clipping
glTranslatef(x,y,z) glRotatef(th,x,y,z) ....
gluLookAt(...)
/ 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)
NDCS
device
DCS
26
Coordinate Systems
http//www.btinternet.com/danbgs/perspective/
27
Perspective Derivation
VCS
NDCS
ytop
y
xleft
(1,1,1)
y
z
(-1,-1,-1)
x
z
z-near
ybottom
z-far
x
xright
28
Perspective Derivation
earlier
complete shear, scale, projection-normalization
29
Perspective Derivation
30
Perspective Derivation

• similarly for other 5 planes
• 6 planes, 6 unknowns

31
Perspective Example

• view volume
• left -1, right 1
• bot -1, top 1
• near 1, far 4

32
Perspective Example
/ w
33
Asymmetric Frusta

• our formulation allows asymmetry
• why bother?

x
x
right
right
Frustum
Frustum
-z
-z
left
left
z-n
z-f
34
Simpler Formulation

• left, right, bottom, top, near, far
• nonintuitive
• often overkill
• look through window center
• symmetric frustum
• constraints
• left -right, bottom -top

35
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
36
Perspective OpenGL
glMatrixMode(GL_PROJECTION) glLoadIdentity() gl
Frustum(left,right,bot,top,near,far)
or glPerspective(fovy,aspect,near,far)
37
Demo Frustum vs. FOV

• Nate Robins tutorial (take 2)
• http//www.xmission.com/nate/tutors.html

38
Projection Taxonomy
planar projections
perspective 1,2,3-point
parallel
orthographic
oblique
cavalier
cabinet
axonometric isometric dimetric trimetric
top, front, side
http//ceprofs.tamu.edu/tkramer/ENGR20111/5.1/20
39
Perspective Projections

• classified by vanishing points

two-point perspective
three-point perspective
40
Parallel Projection

• projectors are all parallel
• vs. perspective projectors that converge
• orthographic projectors perpendicular to
  projection plane
• oblique projectors not necessarily perpendicular
  to projection plane

Oblique
Orthographic
41
Axonometric Projections

• projectors perpendicular to image plane
• select axis lengths

http//ceprofs.tamu.edu/tkramer/ENGR20111/5.1/20
42
Oblique Projections

• projectors oblique to image plane
• select angle between front and z axis
• lengths remain constant
• both have true front view
• cavalier distance true
• cabinet distance half

d / 2
y
y
d
d
d
x
z
x
z
cabinet
cavalier
43
Demos

• Tuebingen applets from Frank Hanisch
• http//www.gris.uni-tuebingen.de/projects/grdev/do
  c/html/etc/AppletIndex.htmlTransformationen