transformations ii

Transformations II

CS5600 Computer Graphics

Rich Riesenfeld

Spring 2005

Lect

ure

Set

7

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Arbitrary 3D Rotation

• What is its inverse?

• What is its transpose?

• Can we constructively elucidate this

relationship?

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Want to rotate about arbitrary axis a

a

)(: Ra

x

z

y

3

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First rotate about z by

( ): Rz

a Now in the

(y-z)-plane

x

z

y

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Then rotate about x by

x

z

y

( ): Rx

Rotate in the

(y-z)-plane

a

)(: Raxisz zNow perform rotation about

x

z

y

aNow a-axis aligned

with z-axis

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Then rotate about x by ( ): Rx

Rotate again in the (y-z)-plane

x

z

y

a

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Then rotate about z by ( ): Rz

Now to original position of a

a

x

z

y

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We effected a rotation by about arbitrary axis a

a

)(: Ra

x

z

y

9

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

10

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

)()( RRx z

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Rotation about arbitrary axis a

• 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

cos( ) sin( ) 0 0 1 0 0 0

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

0 0 1 0 0 sin( ) cos( ) 0

0 0 0 1 0 0 0 1

( )Ra

1000010000cossin00sincos

1000

0cossin0

0sincos00001

1000010000cossin00sincos

)( Rz

In matrix terms, )( Rz

)( Rx

)( Rx

)( Rz

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cos( ) sin( ) 0 0 1 0 0 0

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

0 0 1 0 0 sin( ) cos( ) 0

0 0 0 1 0 0 0 1

( )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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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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Differert Perspectives 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 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

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

1 1

1

1 0 0

0 1 0

0 1 1px

x

xp p

pyx p

x

p

pp

x x

y y

x

y

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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) hand drawing was the first extant complex geometrical form rendered according to the laws of linear perspective

Perspective Study of a Chalice, Drawing, Gabinetto dei Disegni, Uffizi, Florence, ca 1430)

65

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Perspective in Cubism

Woman with a Guitar (1913) G

eorg

es B

raqu

e

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Perspective in Cubism

Madre con niño muerto (1937)

68

Pablo P

icaso

Pablo Picaso, Cabeza de mujer llorando con pañuelo

69

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Perspective (Mural) Games

M C Esher, Another World II

(1947)

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PerspectiveAscending and Descending (1960)

M C

Escher

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M. C. Escher

M C Escher, Ascending and Descending (1960)

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M C Escher

• Perspective is “local”• Perspective consistency is not

“transitive”

• Nonplanar (hyperbolic)

projection

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Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and Hell

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Nonplanar (Hyperbolic) Projection

M C Esher, Heaven and

Hell

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David McAllister

The March of Progress,

(1995)

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Joan Miro: Flat Perspective

The Tilled Field

What cues are missing?

Henri Matisse, La Lecon de

Musique

Flat Perspective: What cues are

missing?

78

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Next 2 Images Contain Nudity !

Henri Matisse, Danse (1909)80

Henri Matisse, Danse II (1910)81

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Atlas Projection

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Norway is at High Latitude

There is considerable size distortion

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Isometric View

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Engineering Drawing: 2 Planes

AA

AA

Section AA

Engineering Drawing: Exploded

View

Understanding 3D Assembly

in a 2D Medium 86

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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

1

1

1

1 0 0

0 1 0

0 1 1 xp p

pyx p

pxx p

x pp

px

x p

py

x p

x x

y y

x

y

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Geometry is Same for Eye at Originy

x

(x,y)

p

h

Screen Plane

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What Happens to Special Points?

What is this point??

1

1 0 0

0 1 0

0 1 1 0

0 0

p

p p

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Let’s Look at Limit

1

1lim 0 0

01

n

nn

n

We see that

Observe,

on -axis0

nx

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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 ?

1 1

1 11 0 0

0 1 0

0 1 10

0 0 00

p p

p p

It comes back to virtual eye point!

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What Does This Mean?

x

y

p

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What Does This Mean?y

p

x

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The “Pencil of Lines” Becomes Parallel

y

x

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Parallel Lines Become “Pencil of Lines” !

x

y

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Parallel Lines Become “Pencil of Lines” !

x

y

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What Does This Mean?

x

y

p

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“True” Perspective in 2Dy

p

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“True” Perspective in 2Dy

p

p

p

p

p

p

p

p

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Viewing Frustum

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What happens for large p?”

1 0 01 0 0

0 1 0 0 1 0

0 1 0 1

1

1 01 1

lim 0

p

p

x x

y y

p

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Projection Becomes Orthogonal: “Right Thing Happens”

x

(x,y)

h=y

p

The End

Transformations II

Lect

ure

Set

7