Assignment #3

As a graphics engineer, can you implement

a “skeletal” rendering pipeline?

Computer Graphics

graphics

primitives

modeling

transform

viewing

transformclipping

shading &

texture

projectionimages in

Internal buffer

transform

matrix

Eye, lookat,

headup

material,

lights,

surface color

Parallel or

Persepctive

volume

images on

screenviewport

transform

viewport

location

Computer Graphics

glMatrixMode(GL_MODELVIEW);

glLoadIdentity(T1);

glMultiMatrixf(T2);

…

glMultiMatrixf(Tn);

draw_the_object(v);

v’ = IT1T2Tnv

Modeling Transform

As a global system

objects move but coordinates stay the same

apply in the reverse order

Tasks

Need a stack (CS16, 24)

Push & pop

Need to create matrices (4x4)

Need to multiply matrices (4x4)

Computer Graphics

Why 4x4 not 3x3?

Use of homogeneous coordinates

[x,y] -> [wx,wy,w]

[wx,wy,w] -> [wx/w, wy/w, w] -> [x,y]

[x,y,z] -> [wx, wy, wz, w]

[wx, wy, wz, w] -> [wx/w, wy/w, wz/w] -

>[x,y,z]

One dimension up (w!=0)

Computer Graphics

Reason #1

All operations (including translation) are

now matrix operations

Hierarchical transforms (multiple T, R, S)

are computed once (top matrix in the stack)

and applied

Computer Graphics

Computer Graphics

Euler Angle Rotation

X Y

Z

XY

Z

X

Y

Z

x

y

z

x

y

z

'

'

'

cos sin

sin cos

1

0 0

0 0

0 0 1 0

0 0 0 1 1

x

y

z

x

y

z

'

'

'

cos sin

sin cos

1

1 0 0 0

0 0

0 0

0 0 0 1 1

11000

0cos0sin

0010

0sin0cos

1

'

'

'

z

y

x

z

y

x

Transformations

Translation Scaling

X

Y

Zz

x

y

X

Y

Z 2x

11000

100

010

001

1

'

'

'

z

y

x

T

T

T

z

y

x

z

y

x

11000

000

000

000

1

'

'

'

z

y

x

S

S

S

z

y

x

z

y

x

Modeling Transform Your Way

Implement a stack with push and pop

Replace OpenGL codes with your own

codes

Computer Graphics

Computer Graphics

Viewing Transform

So the user specifies a lot of information

Eye

Center

Up

Near, far,

Left, right top, bottom, etc.

X

Y

Z

UP

EYE

rightleft

bottom

top

bf

CENTER

Computer Graphics

What does a system programmer do with those numbers?

Generate screen coordinates correctly and efficiently

Inside/outside test

Projection

Here comes the part which contains math which you may not like

But all you need to know is matrix operation

What does OpenGL do?

Computer Graphics

Arbitrary View Volume

X

Y

Z

UP

EYE

rightleft

bottom

top

bf

CENTER

Computer Graphics

Inside-Outside Test (Clipping)

Intersection of

A plane and

A Line

line of points end:),,(),,,(

10

)()()(

0)]([)]([)]([

)(

)(

)(

:

0:

222111

121212

111

121121121

121

121

121

zyxzyx

t

zzcyybxxa

dczbyaxt

dzztzcyytybxxtxa

zztz

yyty

xxtx

line

dczbyaxplane

EYE

Computer Graphics

Clipping in Canonical Volumes

Z

Y

Y=-Z

Y=Z

Z=-1

Z=Zmin

45^o

Z

Y Y=1

Y=-1

Z=-1

(A,B,C)(A,B,C)

far plane=film

near plane

Computer Graphics

Clipping with 6-bit outcode

Perspective

Above y>-z

Below y<z

Right x>-z

Left x<z

Behind z<-1

In front z >zmin

Parallel

Above y>1

Below y<-1

Right x>1

Left x<-1

Behind z<-1

In front z >0

Computer Graphics

Projection

Again, an intersection of

A plane and

A Line

EYE

Computer Graphics

Sidebar: Reason #2

A pin-hole model without inversion (f=1 in

OpenGL)

f (v)

Z

X

x fX

Z

y fY

Z

f (v) x

x

Computer Graphics

Canonical Volumes

Z

Y

Y=-Z

Y=Z

Z=-1

Z=Zmin

45^o

Z

Y Y=1

Y=-1

Z=-1

(A,B,C) (A,B) (A,B,C)(A/C,B/C)

far plane=film

near plane

Sidebar: Reason #2

Traditional Way Matrix Way

Computer Graphics

x fX

Z

y fY

Z

10100

1000

0010

0001

1

'

'

'

z

y

x

z

y

x

Computer Graphics

Problem

Both clipping and projection can be done

efficiently in a canonical volume

But we do not have a canonical volume in

general

Solution: Normalization transform

A single matrix operation to bring objects in

any arbitrary volume into a canonical volume

Cannot change what the user sees

Computer Graphics

Normalization Transform A transformation to facilitate clipping and

projection

X

Y

Z

An arbitrary view volume:

Expensive for clipping and projection

Computer Graphics

X

Y

Z

The canonical view volume:

Simple clipping (six-bit outcode)

Simple projection (x/z, y/z)

Y

Z

y z

y z

z 1

z z min

45o

Computer Graphics

OpenGL Terminology

X

Y

Z

UP

EYE

rightleft

bottom

top

bf

CENTER

Computer Graphics

Normalization TransformPerspective - OpenGL

External parameters

Translate EYE into origin

Rotate the EYE coordinate system such thatw (e-c) becomes z

u becomes x

v becomes y

Internal parameters

Shear to have centerline of the view volume

aligning with z

Scale into canonical truncated pyramid

Computer Graphics

Existing Rendering Pipeline

graphics

primitives

modeling

transform

viewing

transformclipping

shading &

texture

projectionimages in

Internal buffer

transform

matrix

Eye, lookat,

headup

material,

lights,

surface color

Parallel or

Perspective

volume

images on

screenviewport

transform

viewport

location

Computer Graphics

Rendering Pipeline with

Normalization Transform

graphics

primitives

modeling

transform

viewing

transformclipping

shading &

texture

projectionimages in

Internal buffer

transform

matrix

Eye, lookat,

headup

material,

lights,

surface color

Parallel or

Perspective

volume

images on

screenviewport

transform

viewport

location

normalization

transform

Computer Graphics

Changes

Modeling + Viewing + Normalization get

concatenated into ONE transform before

applying to any primitives

Confusion: normalization does not just push

the eye frame back to origin and line up

with world frame, it pushes objects away

too

Purpose: to make clipping and projection

much more efficient

Clipping?

Polygon clipping algorithm discussed

earlier

For inside/outside determination

What Else?

Start from polygon with vertices

End with polygon with vertices

Visible surface determination

Painter’s algorithm (sort by depth)

2D painting

Interior: Scan conversion

Exterior: Line drawing (Bresemheim)

That is!

Computer Graphics

Computer Graphics

Viewing Normalization

Line up (X-Y-Z) and (U-V-W)

Initially, (U-V-W) are specified in (X-Y-Z)

system (In fact, everything is specified in

X-Y-Z system)

Some point in time, want to specify things

in (U-V-W) system, or U becomes (1,0,0),

V becomes (0,1,0), W becomes (0,0,1)

Translation (easy) + Rotation (hard)

Hand-eye Coordination A common problem in graphics, robotics

and vision

The camera and effector are not co-located

Computer Graphics

From camera to robot

R and O are in camera’s frame

Rows are robot’s frame in

camera’s system

Computer Graphics

x

yz

o

P=(xc,yc,zc)

P’=(xr,yr,zr)?

)(

'

'

''

OP

z

y

x

P

Computer Graphics

X YZ

o

P=(xc,yc,zc)

P’=(xr,yr,zr)?

TPOPOPOP

z

y

x

P

|||

|||

|||

|||

|||

|||

)(

|||

|||

)(

'

'

'' ZYXZYXZYXZYX

T

From robot to camera

R and T are in robot’s frame

Cols are camera’s frame in robot’s system

Computer Graphics

Translate EYE into the origin

X

Y

Z

u

v

w

X

Y

Z

u

v

w

1000

100

010

001

1

z

y

x

EYE

EYE

EYE

T

Computer Graphics

Viewing Normalization

Three rotations

Rotate about Y

Rotate about X

Rotate about Z

Computer Graphics

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Y rotation

X rotation

Z rotation

UP

E-C

Computer Graphics

Viewing Normalization

we c

e c

uup w

up w

v w u

| |

| |

u

v

w

u u u

v v v

w w w

x

y

z

x y z

x y z

x y z

1

0

0

0

0 0 0 1 1

Figuring out [u, v, w] in [x, y, z] system

Applying a rotation to transform [x, y, z]

coordinates into [u, v, w] coordinates

Computer Graphics

Rotate EYE coordinate to align w. world system

X

YZ

u

v

wX

Y

Z

u

v

w

Computer Graphics

Shear

X

Y

Z

u

v

wX

Y

Z

u

v

),2

,2

( nearbottomtoprightleft

),0,0( near

near

bottomtop

bnear

rightleft

a

b

a

SH

2,2

,

1000

0100

010

001

w

Computer Graphics

Scale into canonical volume

-z

Y2

bottomtop

near far

-z

Y -1

45o

)1,

2

,

2

(1 bottomtop

near

leftright

nearS

)1

,1

,1

(2farfarfar

S

• scale in x and y

• scale in z

2

bottomtop

Computer Graphics

Example

101

)0,20(),()0,20(),(

)0,1,0()0,0,0(

)10,10,10(

BF

bottomtopleftright

UPCENTEREYE

X

Y

Z

Computer Graphics

• Translate EYE into the origin

T1

1 0 0 10

0 1 0 10

0 0 1 10

0 0 0 1

Rotate EYE to align with the world system

R

1

20

1

20

1

6

2

6

1

60

1

3

1

3

1

30

0 0 0 1

)1,2,1(6

1)1,0,1()1,1,1(

6

12

)1,0,1(

|)1,1,1()0,1,0(|

)1,1,1()0,1,0(

||

3

)1,1,1(

||

uwv

wUP

wUPu

ce

cew

Computer Graphics

• Shear

102,102

,

1000

0100

01010

01001

near

bottomtop

bnear

rightleft

a

SH

Computer Graphics

• Scale into canonical volume

• scale in x and y

• scale in z

1000

0100

0010

10

00010

1

1S

1000

010

100

0010

10

00010

1

1S

Comparing Canonical Volumes

One additional division is needed in perspective

Z

Y

Y=-Z

Y=Z

Z=-1

Z=Zmin

45^o

Z

Y Y=1

Y=-1

Z=-1

(A,B,C) (A,B) (A,B,C)(A/C,B/C)

far plane=film

near plane

(1,-1)

(-1,-1)

(1,-1)

(-1,-1)

Matrix Magic

Try this:

Map perspective volume into parallel volume to

save the division (note Zmin is NEGATIVE)

Computer Graphics

1

0

1

0

0

0

1

0

0100

11

100

0010

0001

1

0

1

0

0

0

1

0

0100

11

100

0010

0001

0100

11

100

0010

0001

min

min

min

min

min

min

min

min

min

min

min

min

min

min

'

min

min

min

z

z

z

z

z

z

z

z

z

z

z

z

z

z

MNN

z

z

z

M

perper

Computer Graphics

Normalization TransformParallel (othographic) - OpenGL

External parameters

Translate EYE into origin

Even though eye is not really where the viewer is

Rotate the EYE coordinate system such that w (e-c) becomes z

u becomes x

v becomes y

Internal parameters

Translate to have centerline of the view volume

aligning with z, and near plane at z=0

Scale into canonical rectangular piped

Computer Graphics

Viewing Normalization

Line up (X-Y-Z) and (U-V-W)

Initially, (U-V-W) are specified in (X-Y-Z)

system (In fact, everything is specified in

X-Y-Z system)

Some point in time, want to specify things

in (U-V-W) system, or U becomes (1,0,0),

V becomes (0,1,0), W becomes (0,0,1)

Translation (easy) + Rotation (hard)

Computer Graphics

Translate EYE into the origin

X

Y

Z

u

v

w

X

Y

Z

u

v

w

1000

100

010

001

1

z

y

x

EYE

EYE

EYE

T

Computer Graphics

Viewing Normalization

Three rotations

Rotate about Y

Rotate about X

Rotate about Z

Computer Graphics

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Y rotation

X rotation

Z rotation

UP

E-C

Computer Graphics

Viewing Normalization

we c

e c

uup w

up w

v w u

| |

| |

u

v

w

u u u

v v v

w w w

x

y

z

x y z

x y z

x y z

1

0

0

0

0 0 0 1 1

Figuring out [u, v, w] in [x, y, z] system

Applying a rotation to transform [x, y, z]

coordinates into [u, v, w] coordinates

Computer Graphics

Rotate EYE coordinate to align w. world system

X

YZ

u

v

wX

Y

Z

u

v

w

Computer Graphics

Translation

X

Y

Z

u

v

wX

Y

Z

u

v

),2

,2

( nearbottomtoprightleft

)0,0,0(

nearcbottomtop

brightleft

a

c

b

a

T

,2

,2

,

1000

100

010

001

w

Computer Graphics

Scale into canonical volume

-z

Y 2

bottomtop

near Far-near

)1

,

2

1,

2

1(

nearfarbottomtopleftrightS

• scale in x, y, and z

2

bottomtop

-z

Y 1

near -1

1

Computer Graphics

Example

101

)0,20(),()0,20(),(

)0,1,0()0,0,0(

)10,10,10(

BF

bottomtopleftright

UPCENTEREYE

X

Y

Z

Computer Graphics

• Translate EYE into the origin

T1

1 0 0 10

0 1 0 10

0 0 1 10

0 0 0 1

Rotate EYE to align with the world system

R

1

20

1

20

1

6

2

6

1

60

1

3

1

3

1

30

0 0 0 1

)1,2,1(6

1)1,0,1()1,1,1(

6

12

)1,0,1(

|)1,1,1()0,1,0(|

)1,1,1()0,1,0(

||

3

)1,1,1(

||

uwv

wUP

wUPu

ce

cew

Computer Graphics

• Translation

102,102

,

1000

1000

10010

10001

near

bottomtop

bnear

rightleft

a

SH

Computer Graphics

• Scale into canonical volume

• scale in x, y, and z

1000

09

100

0010

10

00010

1

1S