Transformations I

CS5600Computer Graphicsby

Rich Riesenfeld

27 February 2002Lect

ure

Set

5

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Transformations and Matrices

• Transformations are functions

• Matrices are functions representations

• Matrices represent linear transf’s

• s'TransfLinear 2Matrices22 Dx

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What is a 2D Linear Transf ?

Recall from Linear Algebra:

. and vectorsand ascalar for

,)()()(:

yx

yTxaTyxaTDefinition

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Example: Scale in x

),2(),2(),(2

:say 2,by in x, Scale

11001010 yxyxyyxx

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Example: Scale in x by 2

What is the graphical view?

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), 00( yx ), 002( yx

), 11( yx ), 112( yx

Scale in x by 2y

x

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x

y

), 002( yx

), 112( yx

yyxx 1010 ,22

yyxx 1010 ,22

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y

), 00( yx

), 11( yx

yyxx 1010 ),(2

yyxx 1010 ),(

yyxx 1010 ),(2

x

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y

yyxx 1010 ),(2

yyxx 1010 ),(

yyxx 1010 ),(2 y

x

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Summary on Scale

• “Scale then add,” is same as

• “Add then scale”

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Matrix Representation

y

x

y

x 2

10

02

Scale in x by 2:

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Matrix Representation

y

x

y

x

220

01

Scale in y by 2:

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Matrix Representation

y

x

y

x

y

x2

2

2

20

02

Overall Scale by 2:

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Matrix RepresentationShowing Same

)(

22

)(

)(2

10

02

10

10

10

10

10

10

yy

xx

yy

xx

yy

xx

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What about Rotation?

Is it linear?

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Rotate by

x

y

)0,1(

)1,0(

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sin

Rotate by : 1st Quadrant

x

y

)0,1(

)sin,(cos

cos

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Rotate by : 1st Quadrant

)sin,(cos)0,1(

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Rotate by : 2nd Quadrant

x

y

)1,0(

)0,1(

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Rotate by : 2nd Quadrant

x

y

cos

sin

)1,0(

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Rotate by : 2nd Quadrant

)cos,sin()1,0(

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Summary of Rotation by

)sin,(cos)0,1(

)cos,sin()1,0(

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Summary (Column Form)

cos

sin

1

0

sin

cos

0

1

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Using Matrix Notation

sin

cos

0

1

cossin

sin-cos

cos

sin-

1

0

cossin

sin-cos

(Note that unit vectors simply copy columns)

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General Rotation by Matrix

cossin

ysin-cos

cossin

sin-cos

yx

x

y

x

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Who had linear algebra?

Who understand matrices?

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What do the off diagonal elements do?

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Off Diagonal Elements

ybx

x

y

x

b 1

01

y

ayx

y

xa

10

1

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Example 1

y

yx

x

y

xyxT

4.0

14.0

01),(

)1,0(

)0,1(

)1,1(

)0,0(

S

x

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Example 1

x

y

yx

x

yxT

4.0

),(

S

)1,0(

)0,1(

)1,1(

)0,0(

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Example 1

y

)1,0(

)4.0,1(

)0,0(

T(S)

yx

x

yxT

4.0

),()4.1,1(

x

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Example 2

y

y

yx

y

xyxT

6.0

10

6.01),(

S

x

)1,0(

)0,1(

)1,1(

)0,0(

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Example 2

x

y

y

yx

yxT

6.0

),(

S

)1,0(

)0,1(

)1,1(

)0,0(

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Example 2

x

y

)1,0(

)0,1()0,0(

T(S)

)1,6.0(

y

yxyxT

6.0 ),(

)1,6.1(

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Summary

Shear in x:

Shear in y:

y

ayx

y

xaShx 10

1

ybx

x

y

x

bShy 1

01

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Double Shear

ab)(10

1

1

01

1

1

b

aa

b

1

1 ab)(

1

01

10

1

b

a

b

a

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Sample Points: unit inverses

0

11

1

01

bb

1

0

10

1

1

aa

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Geometric View of Shear in x

)0,1(

)1,( a )1,0(

)0,1(

)1,1(

Another Geometric View of Shear in x

x39x x

yy

Another Geometric View of Shear in x

40

x

y

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Geometric View of Shear in y

)1,0(

)0,0(

),1( b

)1,0(

)0,1()0,0(

)1,1(

Another Geometric View of Shear in y

h h42

x

y

x

y

Another Geometric View of Shear in y

43

x

y

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“Lazy 1”

100

010

01 0

11

y

x

y

x

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Translation in x

1

y

x

10 10

010

01

yxdx

xd

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Translation in x

1

y

x

10

0

10

10

01

ydy

x

d y

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Homogeneous Coordinates

100

010

01 0

11

y

x

y

x

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Homogeneous Coordinates

0for ,

y

xy

x

1

y

x

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Homogeneous Coordinates

Homogeneous term effects overall scaling

0,For

y

xy

x

y

x

y

x

1100

010

01

11

0

Homogeneous CoordinatesAn infinite number of points correspond to (x,y,1).

They constitute the whole line (tx,ty,t).

x

w

y

w = 1

(tx,ty,t)

(x,y,1)

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We’ve got Affine Transformations

Linear + Translation

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Compound Transformations

• Build up compound transformations by concatenating elementary ones

• Use for complicated motion

• Use for complicated modeling

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Elementary Transformations

• Scale

• Rotate

• Translate

• Shear

• (Reflect)

)(R)(),( dT ydT x

Rf yRf x,

)(),( dSh ydShx

)(),( S yS x

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Refection about y-axis

0

1

0

1

10

01

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Reflection about y-axis

x

y

),( yx),( yx

)0,1()0,1(

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Reflection about x-axis

1

0

1

0

1-0

01

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Reflection about x-axis

x

y

)1,0( ),( yx

),( yx )1,0(

Is Reflection “Elementary?”

• Can we effect reflection in an elementary way?

• (More elementary means scale, shear, rotation, translation.)

58

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Reflection is Scale (-1)

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Example:Move clock hands

x

),( ba

y

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Example:Move clock hands

),( ba

x

y

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Example:Move clock hands

),( ba

x

y

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Example:Move clock hands

y

),( ba

x

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Clock Transformations

• Translate to Origin

• Move hand with rotation

• Move hand back to clock

• Do other hand

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Clock Transformations

15

),(12),(

),()(),(

)(

twhere

baTtRbaTT

baTtRbaTT

b

s

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Clock Transformations

1

y

x

1

10

01

11

10

01

0000

0)cos()sin(

0)sin()cos(

00

b

a

tt

tt

b

a

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Map [a,b] [0,1]

0[ ]

x

a b

x

1

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Map [a,b] [0,1]

• Translate to Origin

• Map x to translated interval

axx

ababaaba ,0,,

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Map [a,b] [0,1]

• Normalize the interval

• Map x to normalized interval

abaxx

1,0,1

,0

abaaab

ab

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Map [a,b] [0,1])( aT x

abS x1

1100

010

01

y

xa

1

yab

ax

100

010

001

ab

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Just Look at

10

01

ab

1 RThis is a homogeneous form for 1D

)( aT x

abS x1

10

1 a

1

x

1

ab

ax

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Map [a,b] [-1,1]

0[ ]

-1 +1x

a bx

2

ba

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Map [a,b] [-1,1]

• Translate center of interval to origin

• Normalize interval to [-1,1]

2

baxx

22 2

1 bax

bax

ab

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Map [a,b] [-1,1]

• Substitute x =a (analogous for x =b) :

12

)(2

2

22

22

22

ab

ab

baa

ab

baa

ab

x

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Now Map [a,b] [c,d]

• First map [a,b] to [0,1]

– (We already did this)

• Then map [0,1] to [c,d]

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Now Map [a,b] [c,d]

• Scale [0,1] by (d-c)

• Then translate by c

• That is, in 1D homogeneous form:

1)(

110

0

10

1 cxcdxcdc

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All Together: [a,b] [c,d]

110

1

10

01

10

0

10

1 xaab

cdc

110

1

10

010

1 xaab

cdc

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Now Map Rectangles

),(minmin yx

),(maxmax vu

),(minmin vu

),(maxmax yx

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Transformation in x and y

yyvv

xxuu

yx

vu

xx

y

x

y

x

minmax

minmax

minmax

minmax

min

min

min

min

,where

1100

10

01

100

00

00

100

10

01

,

This is the Viewport Transformation

• Good for mapping objects from one

coordinate system to another

• This is what we do with windows and

viewports

80

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3D Transformations

• Scale

• Rotate

• Translate

• Shear

)(),(),( dT zdT ydT x

)(),(),( dShzdSh ydShx

)(),(),( S zS yS x

)(),(),( RzR yRx

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3D Scale in x

1000

0100

0010

000

)(

S x

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3D Scale in x

111000

0100

0010

000

z

y

x

z

y

x

S x

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3D Scale in y

111000

0100

000

0001

)(z

y

x

z

y

x

S y

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3D Scale in z

111000

000

0010

0001

)(z

y

x

z

y

x

S z

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Overall 3D Scale

11 1000

0100

0010

0001

)(z

y

x

z

y

x

S

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Overall 3D Scale

Same in x,y and z:

z

y

x

z

y

x

z

y

x

11

What is a Positive Rotation in 3D ?

• Sit at end of given axis

• Look at Origin

• CC Rotation is in Positive direction

88

3D Positive Rotations

x

z

y

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3D Rotation about z-axis by

We have already done this:

11000

0100

00cossin

00sincos

)(z

y

x

Rz

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3D Rotation about x-axis by

y

z

)0,1,0(

)1,0,0(

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3D Rotation about x-axis

11000

0cossin0

0sincos0

0001

)(z

y

x

Rx

3D Rotation about y-axis by

x

z

)0,0,1(

)1,0,0(

93

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3D Rotation about y-axis

11000

0cos0sin

0010

0sin0cos

)(z

y

x

R y

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Elementary Transformations

• Scale

• Rotate

• Translate

• Shear

• (Reflect)

)(),( R yRx)(),( dT ydT x

)(),( S yS x

Rf yRf x,

)(),( dSh ydShx

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Consider an arbitrary 3D rotation

• What is its inverse?

• What is its transpose?

• Can we constructively elucidate this

relationship?

Want to rotate by about arbitrary axis a

a

)(: Ra

x

z

y

97

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First rotate about by

)(: Rzz

a Now in the

(y-z)-plane

x

z

y

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Then rotate about by

x

z

y

)(: Rx x

Rotate in the

(y-z)-plane

a

Now perform rotation about -

x

z

y

)(: Raxisz za Now aligned

with z-axis

100

Now perform rotation about -

x

z

y

)(: Raxisz zaNow aligned

with z-axis

101

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Then rotate about by )(: Rx x

Rotate again in the (y-z)-plane

x

z

y

a

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Then rotate about by )(: Rz z

Now to original position of a

a

x

z

y

We effected a rotation by about arbitrary axis a

a

)(: Ra

x

z

y

104

We effected a rotation by about arbitrary axis a )(: Ra

105

)()()( RRR z xa)( Rz

)()( RRx z

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Rotation about an arbitrary axis

• Rotation about a-axis can be effected by a composition of 5 elementary rotations

• We show arbitrary rotation as succession of 5 rotations about principal axes

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)( Ra

1000

0)cos()sin(0

0)sin()cos(00001

10000100

00)cos()sin(

00)sin()cos(

)(

Ra

1000010000cossin00sincos

1000

0cossin0

0sincos00001

1000010000cossin00sincos

)( Rz

In matrix terms, )( Rz

)( Rx

)( Rx

)( Rz

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1000

0)cos()sin(0

0)sin()cos(00001

10000100

00)cos()sin(

00)sin()cos(

)(

Ra

,)()(1 RaRa

1000010000cossin00sincos

1000

0cossin0

0sincos00001

1000010000cossin00sincos

)( Rz

Similarly, so,

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Recall, tAtBtAB

RtMtRtA

tttt RRMt

RMR .

Consequently, for , RMtRA

because,

RMR tt

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RStMtStRt

RSMtStR

It follows directly that,

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)()(1 RtaRa

)( Rtz

1000

0)cos()sin(0

0)sin()cos(00001

10000100

00)cos()sin(

00)sin()cos(

)(

Ra

1000010000cossin00sincos

1000

0cossin0

0sincos00001

1000010000cossin00sincos

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)()( 1 RtaRa

Constructively, we have shown,

This will be useful later

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3D Translation in x

111000

0100

0010

001

)(z

yd xx

z

y

xd x

d xT x

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3D Translation in y

111000

0100

010

0001

)(zd yy

x

z

y

x

d yd yT y

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3D Translation in z

111000

100

0010

0001

)(d zz

y

x

z

y

x

d zd zT z

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3D Shear in x -direction

111000

0100

0010

001

)(z

y

ayx

z

y

xa

aShx

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3D Shear in x -direction

111000

0100

0010

001

)(z

y

bzx

z

y

xb

bShx

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3D Shears:clamp a principal plane, shear in other 2 DoFs

x

z

y

x

z

y

111000

0100

0010

001

)(z

y

ayx

z

y

xa

aShx

x

z

y

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3D Shear in y -direction

111000

0100

010

0001

)(z

czy

x

z

y

x

ccShy

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3D Shear in y -direction

111000

0100

001

0001

)(z

ydx

x

z

y

x

ddShy

x

z

y

x

z

y

111000

0100

001

0001

)(z

ydx

x

z

y

x

ddShy

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3D Shear in z -direction

111000

010

0010

0001

)(zex

y

x

z

y

x

eeShz

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3D Shear in z

111000

010

0010

0001

)(zex

y

x

z

y

x

eeShz

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3D Shear in z

111000

010

0010

0001

)(zfy

y

x

z

y

x

ffShz

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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

Many Kinds of Perspective Used

• Mechanical Engineering

• Cartography

• Art

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Perspective in Art

• Naïve (wrong)

• Egyptian

• Cubist (unrealistic)

• Esher

– Impossible (exploits local property)

– Hyperpolic (non-planar)

– etc

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“True” Perspective in 2D

y

x

(x,y)

p

h

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“True” Perspective in 2D

pxpyh

pxy

ph

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“True” Perspective in 2D

px

py

px

px

px

py

px

px

p

pxpx

y

x

y

x

11

This is right answer for screen projection

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“True” Perspective in 2D

11101

010

001

yp

x

y

x

p

End Transformations I

Lect

ure

Set

5

133