Post on 21-Jan-2016
Jinxiang Chai
CSCE 441 Computer Graphics3-D Viewing
1
Outline
3D Viewing
Required readings: HB 10-1 to 10-10
Compile and run the codes in page 388
- opengl three-dimensional viewing program example
2
Taking Pictures Using A Real Camera
Steps:
- Identify interesting objects
- Rotate and translate the camera to a desired camera viewpoint
- Adjust camera settings such as focal length
- Choose desired resolution and aspect ratio, etc.
- Take a snapshot
Taking Pictures Using A Real Camera
Steps:
- Identify interesting objects
- Rotate and translate the camera to a desired camera viewpoint
- Adjust camera settings such as focal length
- Choose desired resolution and aspect ratio, etc.
- Take a snapshot
Graphics does the same thing for rendering an image for 3D geometric objects
Image space
3D Geometry Pipeline
5
Normalized
projection space
View spaceWorld spaceObject space
Aspect ratio &
resolution
Focal length
Rotate and translate the
camera
11
z
y
x
MCPVd
v
u
3D Geometry Pipeline
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
6
Model space
(Object space)
3D Geometry Pipeline
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
7
World space
(Object space)
3D Geometry Pipeline
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
8
Eye space
(View space)
3D Geometry Pipeline
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
9
Normalized projection
space
3D Geometry Pipeline
Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations...
10
Image space, window space, raster
space, screen space, device space
3D Geometry Pipeline
11
Normalized project
space
View spaceWorld spaceObject space
Image space
3D Geometry Pipeline
World spaceObject space
glTranslate*(tx,ty,tz)
11000
100
010
001
1
z
y
x
t
t
t
z
y
x
x
y
x
Translate, scale &rotate
3D Geometry Pipeline
Object space
11000
000
000
000
1
z
y
x
s
s
s
z
y
x
z
y
x
glScale*(sx,sy,sz)
World space
Translate, scale &rotate
3D Geometry Pipeline
Object space
glRotate*),,,( zyx vvv
xy
z),( , zyx vvvr
TRRRRRT zyzyz )()()()()(1
Rotate about r by the angle
World space
Translate, scale &rotate
Image space
3D Geometry Pipeline
15
Screen spaceNormalized project
space
View spaceWorld spaceObject space
Now look at how we would compute the world->eye transformation
3D Geometry Pipeline
View spaceWorld space
Now look at how we would compute the world->eye transformation
3D Geometry Pipeline
View spaceWorld space
Rotate&translate
Camera Coordinate
18
Canonical coordinate system
- usually right handed (looking down –z axis)
- convenient for project and clipping
Camera Coordinate
Mapping from world to eye coordinates
- eye position maps to origin
- right vector maps to x axis
- up vector maps to y axis
- back vector maps to z axis
Camera Coordinate
We have the camera in world coordinates We want to transformation T which takes object from world
to camera
Viewing Transformation
wcw
c pTp ,
We have the camera in world coordinates We want to transformation T which takes object from world
to camera
Trick: find T-1 taking object from camera to world
Viewing Transformation
wcw
c pTp ,
ccw
w pTp 1, )(
We have the camera in world coordinates We want to transformation T which takes object from world
to camera
Trick: find T-1 taking object from camera to world
Viewing Transformation
wcw
c pTp ,
ccw
w pTp 1, )(
plhd
okgc
njfb
miea
TT wccw ,1
,
?
Review: 3D Coordinate Trans.
Transform object description from to
pj
i
i
j
o
o
'kji ijk
1
'
'
'
1000
'
'
'
10
0
0
z
y
x
zkkjkik
ykjjjij
xkijiii
z
y
x
TTT
TTT
TTT
),,( 000 zyx
k
'k
poooop
25
Review: 3D Coordinate Trans.
Transform object description from to
pj
i
0
0
1
i
0
1
0
j
o
o
'kji ijk
1
'
'
'
1000
'''
'''
'''
10
0
0
z
y
x
zkji
ykji
xkji
z
y
x
zzz
yyy
xxx),,( 000 zyx
1
0
0
k
'k
poooop
26
Review: 3D Coordinate Trans.
Transform object description from camera to world
27
),,( zyx uuu),,( zyx nnn),,( zyx vvv
10000
0
0
,1
, znvu
ynvu
xnvu
TTzzz
yyy
xxx
wccw
Trick: find T-1 taking object from camera to world
- eye position maps to origin
- back vector maps to z axis
- up vector maps to y axis
- right vector maps to x axis
Viewing Transformation
10000
0
0
,1
, znvu
ynvu
xnvu
TTzzz
yyy
xxx
wccw
Trick: find T-1 taking object from camera to world
Viewing Transformation
1
0
1
0
0
0
, 1000
1000
pnvu
znvu
ynvu
xnvu
Tzzz
yyy
xxx
cw
100
0
0
, pn
pv
pu
n
v
u
TT
T
T
cw
H&B equation (10-4)
Viewing Trans: gluLookAt
30
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
Mapping from world to eye coordinates
Viewing Trans: gluLookAt
),,( zyxref atatatp
),,(0 zyx eyeeyeeyep
),,( zyx upupupV
How to determine ?
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
nvu
,,
Mapping from world to eye coordinates
Viewing Trans: gluLookAt
),,( zyxref atatatp
),,(0 zyx eyeeyeeyep
),,( zyx upupupV
How to determine ?
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
nvu
,,
Mapping from world to eye coordinates
Viewing Trans: gluLookAt
),,( zyxref atatatp
),,(0 zyx eyeeyeeyep
refppN
0
),,( zyx upupupV
N
Nn
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
Mapping from world to eye coordinates
Viewing Trans: gluLookAt
),,( zyxref atatatp
),,(0 zyx eyeeyeeyep
refppN
0
),,( zyx upupupV
N
Nn
V
nVu
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
Mapping from world to eye coordinates
Viewing Trans: gluLookAt
),,( zyxref atatatp
),,(0 zyx eyeeyeeyep
refppN
0
),,( zyx upupupV
N
Nn
V
nVu
unv
H&B equation (10-1)
gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)
Viewing Trans: gluLookAt
Test: gluLookAt( 4.0, 2.0, 1.0, 2.0, 4.0, -3.0, 0, 1.0, 0 )
- What’s the transformation matrix from the world space to the camera reference system?
3D Geometry Pipeline
View spaceWorld space
3D-3D viewing
transformation
Projection
General definition
transform points in n-space to m-space (m<n)
In computer graphics
map 3D coordinates to 2D screen coordinates
Projection
General definition
transform points in n-space to m-space (m<n)
In computer graphics
map 3D coordinates to 2D screen coordinates
How can we project 3d objects to 2d screen space?
How Do We See the World?
Let’s design a camera:
idea 1: put a piece of film in front of camera
Do we get a reasonable picture?
Pin-hole Camera
Add a barrier to block off most of the raysThis reduces blurringThe opening known as the apertureHow does this transform the image?
Camera Obscura
The first camera Known to Aristotle Depth of the room is the focal length Pencil of rays – all rays through a point
Perspective Projection
Maps points onto “view plane” along projectors emanating from “center of projection” (COP)
43
Perspective Projection
Maps points onto “view plane” along projectors emanating from “center of projection” (COP)
44
What’s relationship between 3D points and
projected 2D points?
Consider the projection of a 3D point on the camera plane
3D->2D
45
Consider the projection of a 3D point on the camera plane
3D->2D
46
Consider the projection of a 3D point on the camera plane
3D->2D
47
By similar triangles, we can compute how much the x and y coordinates are scaled
Consider the projection of a 3D point on the camera plane
3D->2D
48
By similar triangles, we can compute how much the x and y coordinates are scaled
z
dxx
d
x
z
x
'
z
dyy
d
y
z
y
'
'
Homogeneous Coordinates
Is this a linear transformation?
Homogeneous Coordinates
Is this a linear transformation?– no—division by z is nonlinear
Homogeneous Coordinates
Is this a linear transformation?
Trick: add one more coordinate:
homogeneous image
coordinates
homogeneous scene
coordinates
– no—division by z is nonlinear
Remember how we said 2D/3D geometric transformations work with the last coordinate always set to one
What happens if the coordinate is not one We divide all coordinates by w:
Homogeneous Point Revisited
52
If w=1, nothing happens
Sometimes, we call this division step the “perspective divide”
Now we can rewrite the perspective projection equation as matrix-vector multiplications
The Perspective Matrix
53
z
dxx
d
x
z
x
'
z
dyy
d
y
z
y
'
'
Now we can rewrite the perspective projection equation as matrix-vector multiplications
The Perspective Matrix
54
z
dxx
d
x
z
x
'
z
dyy
d
y
z
y
'
'
10100
0010
0001
'
'
'
z
y
x
dw
y
x
This becomes a linear transformation!
Now we can rewrite the perspective projection equation as matrix-vector multiplications
The Perspective Matrix
55
After the division by w, we have
z
dxx
d
x
z
x
'
z
dyy
d
y
z
y
'
'
10100
0010
0001
'
'
'
z
y
x
dw
y
x
11
'
'
zdy
zdx
y
x
Perspective Effects
Distant object becomes small
The distortion of items when viewed at an angle (spatial foreshortening)
Perspective Effects
Distant object becomes small
The distortion of items when viewed at an angle (spatial foreshortening)
Perspective Effects
Distant object becomes small
The distortion of items when viewed at an angle (spatial foreshortening)
Perspective projection is an example of projective transformation
- lines maps to lines
- parallel lines do not necessary remain parallel
- ratios are not preserved
Properties of Perspective Proj.
Perspective projection is an example of projective transformation
- lines maps to lines
- parallel lines do not necessary remain parallel
- ratios are not preserved One of advantages of perspective projection is that
size varies inversely proportional to the distance-looks realistic
Properties of Perspective Proj.
What happens to parallel lines they are not parallel to the projection plane?
Vanishing Points
61
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
Vanishing Points
62
01z
y
x
z
y
x
v
v
v
tp
p
p
vtpl
p v
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
After perspective transformation, we have
Vanishing Points
63
01z
y
x
z
y
x
v
v
v
tp
p
p
vtpl
11
'
'
zdy
zdx
y
x
d
tvp
tvp
dtvp
tvp
y
x
zz
yy
zz
xx
11
'
'
p v
Letting t go to infinity:
Vanishing Points (cont.)
64
d
tvp
tvp
dtvp
tvp
y
x
zz
yy
zz
xx
11
'
'
d
v
v
dv
v
y
x
z
y
z
x
11
'
'
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
Vanishing Points
65
01z
y
x
z
y
x
v
v
v
tp
p
p
vtpl
d
v
v
dv
v
y
x
z
y
z
x
11
'
'
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
How about the line
Vanishing Points
66
01z
y
x
z
y
x
v
v
v
tp
p
p
vtpl
d
v
v
dv
v
y
x
z
y
z
x
11
'
'
01z
y
x
z
y
x
v
v
v
tq
q
q
vtql
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
How about the line
Vanishing Points
67
01z
y
x
z
y
x
v
v
v
tp
p
p
vtpl
d
v
v
dv
v
y
x
z
y
z
x
11
'
'
01z
y
x
z
y
x
v
v
v
tq
q
q
vtql
d
v
v
dv
v
y
x
z
y
z
x
11
'
'
What happens to parallel lines they are not parallel to the projection plane?
The equation of the line:
How about the line
Vanishing Points
68
01z
y
x
z
y
x
v
v
v
tp
p
p
vtpl
d
v
v
dv
v
y
x
z
y
z
x
11
'
'
01z
y
x
z
y
x
v
v
v
tq
q
q
vtql
d
v
v
dv
v
y
x
z
y
z
x
11
'
'
Same vanishing point!
What happens to parallel lines they are not parallel to the projection plane?
Each set of parallel lines intersect at a vanishing point on the PP
Vanishing Points
What happens to parallel lines they are not parallel to the projection plane?
Each set of parallel lines intersect at a vanishing point on the PP
Vanishing Points
Parallel Projection
Center of projection is at infinity
Direction of projection (DOP) same for all points
Orthographic Projection
Direction of projection (DOP) perpendicular to view plane
Orthographic Projection
Direction of projection (DOP) perpendicular to view plane
1000
0000
0010
0001
Properties of Parallel Projection
Not realistic looking Good for exact measurement Are actually affine transformation
- parallel lines remain parallel
- ratios are preserved
- angles are often not preserved Most often used in CAD, architectural drawings,
etc. where taking exact measurement is important
Perspective projection from 3D to 2D
3D->2D
75
10100
0010
0001
'
'
'
z
y
x
dw
y
x
Perspective projection from 3D to 2D
3D->2D
76
10100
0010
0001
'
'
'
z
y
x
dw
y
x
But so far, we have not considered the size of film
plane!
We have also not considered visibility problem
Image space
3D Geometry Pipeline
77
Screen spaceNormalized project
space
View spaceWorld spaceObject space
z = -1z = 1
The center of projection and the portion of projection plane that map to the final image form an infinite pyramid. The sides of pyramid called clipping planes
Additional clipping planes are inserted to restrict the range of depths
Perspective Projection Volume
OpenGL Perspective-Projection
79
Normalized project
space
View space
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
gluPerspective(fovy,aspect,dnear, dfar)
z = -1z = 1
View space
(xwmin,ywmin,Znear)(xwmax,ywmax,znear)
General Perspective-Projection
zfar
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
Six parameters define six clipping planes!
nearnear zd
General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
Left-vertical clipping plane
General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
right-vertical clipping plane
General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
bottom-horizontal clipping plane
General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
top-horizontal clipping plane
General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
near clipping plane
nearnear zd
General Perspective-Projection
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
far clipping plane
farfar zd
Normalized project space (left-
handed)
View space (right-
handed)
B’=(1,1,1)
A’=(-1,-1,-1)
General Perspective-Projection
A maps to A’, B maps to B’Keep the directions of x and y axes!
A
B
B’=(1,1,1)
A’=(-1,-1,-1)
General Perspective-Projection
0100
200
02
0
002
minmax
minmax
minmax
minmax
minmax
minmax
farnear
farnear
farnear
farnear
near
near
zz
zz
zz
zzywyw
ywyw
ywyw
zxwxw
xwxw
xwxw
z
T
H&B equation (10-40)
A
B
OpenGL Symmetric Perspective-Projection Function gluPerspective(fovy,aspect,dnear, dfar)
89
Assume z-axis is the centerline of 3D view frustum!
Image space
3D Geometry Pipeline
90
Normalized project
space
View spaceWorld spaceObject space
Image space-.>Image space
Viewport Transformation
91
glViewport(xmin, ymin, width, height)
(xmin,ymin)
Display window
viewport
Image space-.>Image space
Viewport Transformation
92
glViewport(xmin, ymin, width, height)
(xmin,ymin)
- Besides x and y, each pixel has a depth value z, which is stored in depth buffer.
-Depth values will be used for visibility testing
-The color of the pixel is stored in color buffer
93
Image space
Normalized project
space
B’=(1,1,1)
A’=(-1,-1,-1)
Normalized project space
A’’=(xvmin,yvmin,0)
B’’=(xvmax,yvmax,1)
Viewport Transformation
-1<=z<=1
0<=z<=1
Viewport Transformation
94
Image space
Normalized project
space
B’=(1,1,1)
A’=(-1,-1,-1)
Normalized project space
A’’=(xvmin,yvmin,0)
B’’=(xvmax,yvmax,1)
10002
1
2
100
20
20
200
2minmaxminmax
minmaxminmax
yvyvyvyv
xvxvxvxv
T
H&B equation (10-42)
-1<=z<=1
0<=z<=1
Image space
Summary: 3D Geometry Pipeline
95
Normalized project
space
View spaceWorld spaceObject space
Taking Steps Together
96
11
z
y
x
MCPVd
v
u
Image spaceNormalized project
space
View spaceWorld spaceObject space
OpenGL Codes
97