Computer GraphicsComputer GraphicsCS630CS630

Computer GraphicsComputer GraphicsCS630CS630

Lecture 3 – Transformation And Lecture 3 – Transformation And Coordinate SystemsCoordinate Systems

2

What is a Transformation?

• Maps points (x, y) in one coordinate system to points (x', y') in another coordinate system

x' = ax + by + c

y' = dx + ey + f

3

Transformations• Simple transformation

– Translation– Rotation– Scaling

4

Transformations• Deformable transformations

– Shearing– Tapering– Twisting– Etc..

• Issues– Can be combined– Are these operations invertible?

5

Transformations• Why use transformations?

– Position objects in a scene (modeling)– Change the shape of objects– Create multiple copies of objects– Projection for virtual cameras– Animations

6

Classes of Transformation

• Rigid-Body/ Euclidean transformation

• Similarity Transforms• Linear Transforms• Affine Transforms• Projective Transforms

7

Rigid-Body / Euclidean Transforms

• Preserves distances• Preserves angles

TranslationRotation

Rigid / Euclidean

Identity

8

How are Transforms Represented?

x' = ax + by + c

y' = dx + ey + f

x'

y'

a b

d e

c

f=

x

y+

p' = M p + t

9

Translation

z

y

x

z

y

x

t

t

t

= +

z

y

x

tttT zyx ),,(

z

y

x

z

y

x

t

t

t

= +

'

'

'

z

y

x

10

Properties of Translation

v=v)0,0,0(T

=v),,(),,( zyxzyx tttTsssT

=

=v),,(1zyx tttT

v),,(),,( zyxzyx tttTsssT v),,(),,( zyxzyx sssTtttT

v),,( zzyyxx tststsT

v),,( zyx tttT

11

Scaling

zs

ys

xs

z

y

x

z

y

x

'

'

'

z

y

x

zyx

s

s

s

sssS

00

00

00

),,(

Uniform scaling iff zyx sss

12

Rotations (2D)

sin

cos

ry

rx

cos)sin(sin)cos('

sin)sin(cos)cos('

rry

rrx

)sin('

)cos('

ry

rx

cossinsincos)sin(

sinsincoscos)cos(

cossin'

sincos'

yxy

yxx

yx,

',' yx

x

y

13

Rotations 2D• So in matrix notation

y

x

y

x

cossin

sincos'

'

14

Rotations (3D)

100

0cossin

0sincos

)(

cos0sin

010

sin0cos

)(

cossin0

sincos0

001

)(

z

y

x

R

R

R

15

Properties of RotationsIRa )0(

)()()()( aaaa RRRR

)()()( aaa RRR

)()()(1 Taaa RRR

)()()()( abba RRRR order matters!

16

Combining Translation & Rotation

)1,1(T

)45( R

)45( R

)1,1(T

17

Combining Translation & Rotation

Tvv'

RTR

TR

R

vv

vv

vv

''

)(''

'''

vv R'

TR

T

vv

vv

''

'''

18

Homogeneous Coordinates

• Add an extra dimension• in 2D, we use 3 x 3 matrices• In 3D, we use 4 x 4 matrices

• Each point has an extra value, w

x'

y'

z'

w'

=

x

y

z

w

a

e

i

m

b

f

j

n

c

g

k

o

d

h

l

p

p' = M p

19

Homogeneous Coordinates

• Most of the time w = 1, and we can ignore it

x'

y'

z'

1

=

x

y

z

1

a

e

i

0

b

f

j

0

c

g

k

0

d

h

l

1

20

Homogeneous Coordinates

z

y

x

w

Z

Y

X

can be represented as

wherew

Zz

w

Yy

w

Xx ,,

21

Translation Revisited

11000

100

010

001

),,(z

y

x

t

t

t

z

y

x

tttTz

y

x

zyx

22

Rotation & Scaling Revisited

11000

000

000

000

),,(z

y

x

s

s

s

z

y

x

sssSz

y

x

zyx

11000

0cossin0

0sincos0

0001

)(z

y

x

z

y

x

Rx

23

Combining Transformations

vv

vvvv

vvv

vv

M

TRSTRT

RSR

S

'''

''''''

'''

'

where TRSM

24

Transforming Tangents

t

qp

qp

qpt

qpt

M

M

MM

)(

'''

25

Transforming Normals

nnn

nn

nn

tntn

tn

tn

tn

TT

T

TT

TT

T

T

T

MM

M

M

M

M

11'

'

'

'

0'

0''

0

26

Surface Normal• Surface Normal: unit vector that is locally

perpendicular to the surface

27

Why is the Normal important?

• It's used for shading — makes things look 3D!

object color only Diffuse Shading

28

Visualization of Surface Normal

± x = Red± y = Green± z = Blue

29

How do we transform normals?

Object SpaceWorld Space

nOS

nWS

30

Transform the Normal like the Ray?

• translation?• rotation?• isotropic scale?• scale?• reflection?• shear?• perspective?

31

More Normal Visualizations

Incorrect Normal Transformation Correct Normal Transformation

32

Transforming Normals

nnn

nn

nn

tntn

tn

tn

tn

TT

T

TT

TT

T

T

T

MM

M

M

M

M

11'

'

'

'

0'

0''

0

33

Rotations about an arbitrary axis

Rotate by around a unit axis r

r

34

An Alternative View• We can view the rotation around an

arbitrary axis as a set of simpler steps

• We know how to rotate and translate around the world coordinate system

• Can we use this knowledge to perform the rotation?

35

Rotation about an arbitrary axis

• Translate the space so that the origin of the unit vector is on the world origin

• Rotate such that the extremity of the vector now lies in the xz plane (x-axis rotation)

• Rotate such that the point lies in the z-axis (y-axis rotation)

• Perform the rotation around the z-axis• Undo the previous transformations

36

Rotation about an arbitrary axis

• Step 1Rotate x-axis

x

y

z

(a,b,c)

x

(a’,b’,c’)

37

Closer Look at Y-Z Plane

• Need to rotate degrees around the x-axis

y

z

38

Equations for

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

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

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

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

cb

cb

cb

cb

39

Rotation about the Y-axis

• Using the same analysis as before, we need to rotate degrees around the Y-axis

y

z

x

(a’,b’,c’)=Rx () (a,b,c)T

40

Equations for

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

)1,0,0()',','()cos(

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

||)',','()1,0,0(||)sin(

cba

cba

cba

cba

41

Rotation about the Z-axis

• Now, it is aligned with the Z-axis, thus we can simply rotate degrees around the Z-axis.

• Then undo all the transformations we just did

42

Equation summary

c

b

a

TRRRRRT

c

b

a

rot xyzyxaxis )()()()()(

'

'

'

)( 111

43

Deformations

Transformations that do not preserve shape Non-uniform scaling Shearing Tapering Twisting Bending

44

Shearing

11000

01

01

01

1

'

'

'

z

y

x

ss

ss

ss

z

y

x

zyzx

yzyx

xzxy

0

0

0

1

zyzx

yzyx

xz

xy

ss

ss

s

s

x

y

x

y

45

Tapering

11000

0)(00

00)(0

0001

1

'

'

'

z

y

x

xf

xf

z

y

x

46

Twisting

11000

0))(cos(0))(sin(

0010

0))(sin(0))(cos(

1

'

'

'

z

y

x

yy

yy

z

y

x

47

Bending

11000

0)()(0

0)()(0

0001

1

'

'

'

z

y

x

ykyh

ygyf

z

y

x

48

Quick Recap• Computer Graphics is using a

computer to generate an image from a representation.

Model Imagecomputer

49

Modeling• What we have been studying so

far is the mathematics behind the creation and manipulation of the 3D representation of the object.

Model Imagecomputer

50

What have we seen so far?

• Basic representations (point, vector)• Basic operations on points and vectors

(dot product, cross products, etc.)• Transformation – manipulative

operators on the basic representation (translate, rotate, deformations) – 4x4 matrices to “encode” all these.

51

Why do we need this?• In order to generate a picture from

a model, we need to be able to not only specify a model (representation) but also manipulate the model in order to create more interesting images.

52

Overview• The next set of slides will deal with

the other half of the process. • From a model, how do we

generate an image

Model Imagecomputer

53

Scene Description

Scene

LightsCamera ObjectsMaterialsBackground

54

Graphics Pipeline

55

Graphics Pipeline

• Modeling transforms orient the models within a common coordinate frame (world space)

56

Graphics Pipeline

57

Graphics Pipeline

• Maps world space to eye space

• Viewing position is transformed to origin & direction is oriented along some axis (usually z)

58

Graphics Pipeline

• Transform to Normalized Device Coordinates (NDC)

• Portions of the object outside the view volume (view frustum) are removed

59

Graphics Pipeline

• The objects are projected to the 2D image place (screen space)

60

Graphics Pipeline

61

Graphics Pipeline

• Z-buffer - Each pixel remembers the closest object (depth buffer)

62

Graphics Pipeline• Almost every step in the graphics

pipeline involves a change of coordinate system. Transformations are central to understanding 3D computer graphics.

63

IntuitivelyObjectSpace

WorldSpace

CameraSpace +Projection+NDC

Rasterization

64

Coordinate Systems Object coordinates World coordinates Camera coordinates Normalized device coordinates Window coordinates

65

Object CoordinatesConvenient place to model the

object

O

66

World CoordinatesCommon

coordinates for the scene

O

O

W

TSRM wo

67

Positioning Synthetic Camera

What are our “degrees of freedom” in camera positioning?To achieve effective visual simulation, we want:1) the eye point to be in proximity of modeled scene2) the view to be directed toward region of interest, and3) the image plane to have a reasonable “twist”

68

Eye Coordinates

Eyepoint at originu axis toward “right” of image planev axis toward “top” of image planeview direction along negative n axis

69

Transformation to Eye Coordinates

Our task: construct the transformation M that re-expresses world coordinates in the viewer frame

70

Machinery: Changing Orthobases

Suppose you are given an orthobasis u, v, nWhat is the action of the matrix M withrows u, v, and n as below?

71

Applying M to u, v, n

Two equally valid interpretations, depending on reference frame:1: Think of uvn basis as a rigid object in a fixed world spaceThen M “rotates” uvn basis into xyz basis2: Think of a fixed axis triad, with “labels” from xyz spaceThen M “reexpresses” an xyz point p in uvn coords!It is this second interpretation that we use todayto “relabel” world-space geometry with eye space coordinates

72

Positioning Synthetic Camera

Given eyepoint e, basis ˆu, ˆv, ˆnDeduce M that expresses world in eye coordinates:Overlay origins, then change bases:

73

Positioning Synthetic Camera

Check: does M re-express world geometry in eye coordinates?

74

Positioning Synthetic Camera

Camera specification must include:World-space eye position eWorld-space “lookat direction” -n

Are e and -n enough to determine the camera DOFs (degrees of freedom)?

75

Positioning Synthetic Camera

Are e and -n enough to determine the camera DOFs?No. Note that we were not given u and v!(Why not simply require the user to specify them?)

We must also determine u and v, i.e., camera “twist” about n.Typically done by specification of a world-space “up vector”provided by user interface, e.g., using gluLookat(e, c, up)“Twist” constraint: Align v with world up vector (How?)

76

Positioning Synthetic Camera

77

Where are we?

78

What is Projection?

Any operation that reduces dimension (e.g., 3D to 2D)

Orthographic ProjectionPerspective Projection

79

Orthographic Projection• focal point at infinity • rays are parallel and orthogonal to the image

plane

Image

World

F

F

Image

World

I

W

80

Comparison

81

Simple Perspective Camera

• camera looks along z-axis• focal point is the origin• image plane is parallel to xy-plane at

distance d• d is call focal length

82

Similar TrianglesY

Z

[0, d][0, 0]

[Y, Z]

[(d/Z)Y, d]

• Similar situation with x-coordinate• Similar Triangles:

point [x,y,z] projects to [(d/z)x, (d/z)y, d]

83

Projection Matrix Projection using homogeneous coordinates:

– transform [x, y, z] to [(d/z)x, (d/z)y, d]

•2-D image point:•discard third coordinate•apply viewport transformation to

obtain physical pixel coordinates

d 0 0 0

0 d 0 0

0 0 d 0

0 0 1 0

x

y

z

1

dx dy dz z d

zx

d

zy d

Divide by 4th coordinate

(the “w” coordinate)

84

Perspective Projection

85

Perspective Projection

z = 0 not allowed (what happens to points on plane z = 0?)Operation well-defined for all other points

86

Perspective Projection

Matrix formulation using “homogeneous 4-vectors”:

Finally, recover projected point using homogenous convention:Divide by 4th element to convert 4-vector to 3-vector:

87

Camera CoordinatesCoordinate system with the camera

in a convenient pose

1000

nr

vr

ur

zyx

zyx

zyx

cw nnn

vvv

uuu

Mu

v

n

x

y

z

88

View Volume and Normalized Device

Coordinates

89

Normalized Device Coordinates

Device independent coordinatesVisible coordinate usually range from:

11

11

11

z

y

x

90

Perspective ProjectionTaking the camera coordinates to

NDC

z

x

near

91

Perspective Projection

z

x

near'p

p

z

xnearx

z

x

near

x

'

'

92

Perspective Projection + NDC

0100

200

02

0

002

nearfar

nearfar

nearfar

nearfarbottomtop

bottomtop

bottomtop

nearleftright

leftright

leftright

near

M pc

93

Window CoordinatesAdjusting the NDC to fit the window

),( 00 yx is the lower left of the window

94

Window CoordinatesAdjusting the NDC to fit the window

0

0

2)1(

2)1(

yheight

yy

xwidth

xx

ndw

ndw

),( 00 yx is the lower left of the window

width

height

95

Window Coordinates

1000

01002

02

0

200

2

0

0

_

heighty

height

widthx

width

M pw

96

Summary : Object Coordinate to Device

Coordinate• Take your representation (points) and transform it

from Object Space to World Space (Mwo)• Take your World Space point and transform it to

Camera Space (Mcw)• Perform the remapping and projection onto the

image plane in Normalized Device Coordinates (Mw_p Mpc)

• Perform this set of transformations on each point of the polygonal object (M= Mw_pMpcMcwMwo)