Transformations II

CS5600 Computer Graphicsby

Rich Riesenfeld4 March 2002Le

ctur

e S

et 6

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What About Elementary Inverses?

• Scale• Shear• Rotation• Translation

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Scale Inverse

1001

100

100 1

100

100 11

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Shear Inverse

1

1

1

1

00

101

101

00

101

101

bb

aa

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Shear Inverse

101

101

101

101

1

1

bb

aa

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Rotation Inverse

cossin-sincos

cossinsin-cos

)cos(-)sin(-)sin(--)cos(-

cossinsin-cos

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Rotation Inverse

)cos(sin)sincossin(cos

)sincossin(cos)sin(cos22

22

cossin-sincos

cossinsin-cos

1001

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Rotation Inverse

cossin-sincos

cossinsin-cos 1

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Translation Inverse

10010

01

10010

01

10010

01

10010

01

0

0

0

)(

00

xdxd

xdxd

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Translation Inverse

10010

01

10010

01

00

1

xdxd

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Shear in x then in y)1,0(

)0,0(

)1,0(

)0,0(

)1,0(

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

),(1 bab

),( 1a

),( 1 aba

),( 11 baba

),( 11 a

)1,0(

)0,0(

)1,1( b

)1,1(

),1( b

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Shear in y then in x)1,0(

)0,0(

)1,0(

)0,0(

)1,0(

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

),(1 b

),( 1a

),( 1 aba

),( 11 abba

),( 11 a

)1,0(

)0,0(

)1,1( b

)1,1(

)0,1(

),1( b

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Results Are Different

y then x: x then y:

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Want the RHR to Work

jik

ikj

kji

kj

i

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3D Positive Rotations

x

z

y

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Transformations as a Change in Coordinate System

• Useful in many situations

• Use most natural coordination system locally

• Tie things together in a global system

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Example

x

y

1

2

34

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Example

is the transformation that

takes a point in coordinate system

j and converts it to a point in

coordinate system i

M ji

p j)(

p i)(

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Example

•

•

•

pMpji

ji)()(

pMpkj

kj)()(

MMM kjki ji

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Example

•

•

•

)2,4(12 TM

)3,2()2,2(32 TSM

)8.1,7.6()45(43 TRM

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Recall the Following

ABAB 111)(

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Since

•

•

•

)2,4(21 TM

)21,

21()3,2(23 STM

MM ijji

1

)45()8.1,7.6(34 RTM

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Change of Coordinate System

• Describe the old coordinate system in terms of the new one.

x’y

x

y’

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Change of Coordinate System

Move to the new coordinate system and describe the one old.

x

y

x

y Old is a negative rotation of the new.

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What is “Perspective?”

• A mechanism for portraying 3D in 2D

• “True Perspective” corresponds to projection onto a plane

• “True Perspective” corresponds to an ideal camera image

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Many Kinds of Perspective Used

• Mechanical Engineering

• Cartography

• Art

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Perspective in Art

• Naïve (wrong)• Egyptian• Cubist (unrealistic)• Esher• Miro• Matisse

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Egyptian Frontalism

• Head profile

• Body front

• Eyes full

• Rigid style

Uccello's (1392-1475) handdrawing was the first extant complex geometrical form rendered accor-ding to the laws of linear perspective (Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 1430 1440)

29

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Perspective in Cubism

Braque, Georges

Woman with a Guitar

Sorgues, autumn

1913

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Perspective in Cubism

Pablo Picaso, Madre con niño muerto (1937) 32

Pablo Picaso Cabeza de mujer

llorando con

pañuelo33

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Perspective (Mural) Games

M C Esher, Another World II (1947)

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Perspective

M.C. Escher, Ascending

and Descending

(1960)

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M. C. Esher

M.C. Escher, Ascending

and Descending

(1960)

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M. C. Esher

• Perspective is “local”

• Perspective consistency is not “transitive”

• Nonplanar (hyperbolic) projection

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Nonplanar Projection

M C Esher, Heaven and

Hell

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Nonplanar Projection

M C Esher, Heaven and

Hell

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David McAllister

The March of Progress,

(1995)

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Joan MiroThe Tilled

Field

Flat Perspective:

What cues are missing?

Henri Matisse, La Lecon de

Musique

Flat Perspective:What cues are

missing?

42

Henri Matisse, Danse II (1910) 43

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Norway is at High Latitude

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Isometric View

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Engineering Drawing

A

A

Section AA

Engineering Drawing: Exploded

View

Understanding 3D Assembly in a

2D Medium

48

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“True” Perspective in 2Dy

x

(x,y)

p

h

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“True” Perspective in 2D

pxpyh

pxy

ph

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“True” Perspective in 2D

11 110

010001

px

p

yx

yx

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“True” Perspective in 2D

px

pypx

px

pxpy

pxpx

ppx

yx

px

yx

11

This is right answer for screen projection

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Geometry is Same for Eye at Origin

y

x

(x,y)

p

h

Screen Plane

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What Happens to Special Points?

00

10

101010001 pp

p

What is this point??

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Look at a Limit

axis-on 0

0100

1

1

xn

nn

n

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Where does Eye Point Go?

• It gets sent to on x-axis

• Where does on x-axis go?

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What happens to ?

0100

1

001

10010001

11

pp

pp

It comes back to virtual eye point!

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What Does This Mean?

x

p

y

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The “Pencil of Lines” Becomes Parallel

x

y

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Parallel Lines Become a “Pencil of Lines” !

x

y

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What Does This Mean?

x

p

y

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“True” Perspective in 2Dy

p

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“True” Perspective in 2Dy

p

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Viewing Frustum

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What happens for large p?”

1100

010001

110010001

1

yx

yx

p

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Projection Becomes Orthogonal

x

(x,y)

p

h=y

The End of

Transformations II

67

Lect

ure

Set

6