2IV60!3!2D Transformations

7/25/2019 2IV60!3!2D Transformations

1/57

2IV60 Computer Graphics

2D transformations

Jack van Wijk

TU/e


2/57

vervie!" Wh# transformations$

" %asic transformations&

'trans(ation) rotation) sca(in*

" Com+inin* transformations'homo*enous coor,inates) transform- .atrices

" irst 2D) net 1D


3/57

Transformations

ima*e

train

!or(,

!hee(modelling

instantiation

viewing

animation


4/57

Wh# transformation$

" .o,e( of o+jects

!or(, coor,inates& km, mm, etc.

ierarchica( mo,e(s&&human = torso + arm + arm + head + leg + leg

arm = upperarm + lowerarm + hand

"Vie!in*3oom in) move ,ra!in*) etc-

" 4nimation


5/57

Trans(ation

Trans(ate over vector 5tx, ty

x=x+ tx) y=y+ ty

or

x

y

P

P+T

7% 89:&2209222

T

=

=

=

+=

y

x

t

t

y

x

y

x TPP

TPP'

an,);;;

!ith)


6/57

Trans(ation po(#*on

Trans(ate po(#*on&

4pp(# the same operation

on a(( points-

Works a(!a#s) for a((

transformations of

o+jects ,efine, as a set

of points-

x

y

T

7% 89:&2209222


7/57


8/57


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=cha(e !ith factorsxan,sy&

x= sxx) y= syy

or

an,0

0);

;;

!ith)

=

=

=

=

y

xs

s

y

x

y

xPSP

SPP'

=ca(in*

x

y

P

7% 89:&22>922?

x

P

Q

Q


10/57

=ca(in* !ith respect to a point F

=ca(e !ith factorssxan,sy&

Px= sxPx) Py= syPy

With respect to F&

PxFx= sx5PxFx)

PyFy= sy5PyFy

or Px= Fx+ sx5PxFx)

Py= Fy+ sy5PyFy

x

y

P

7% 89:&22>922?

x

P

Q

QF

P

F


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Transformations

" Trans(ate !ith V&

T@ P A V

" =cha(e !ith factors@s#@s&

S@sP

"


12/57

Transformations

" .ess#

" Transformations !ith respect to points&

even more mess#

" o! to com+ine transformations$

7% 892&22?922B


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omo*eneous coor,inates :

" Uniform representation of trans(ation)

rotation) sca(in*

" Uniforme representation of points an,

vectors

" Compact representation of seEuence of

transformations

7% 892&22?922B


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omo*eneous coor,inaten 2

" 4,, etra coor,inate&

P @ 5p )p# )ph or

x@ 5x, y, h

" Cartesian coor,inates& ,ivi,e +# h

x @ 5x/h)y/h" Foints& h @ : 5for the time +ein*)

vectors& h @ 07% 892&22?922B


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6)5;

or

::00:0

0:

:;

;

&nTrans(atio

PTP yx

y

x

tt

y

x

t

t

y

x

=

=

Trans(ation matri

7% 892&22?922B


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5;

or

::000cossin

0sincos

:;

;

&


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=ca(in* matri

7% 892&22?922B

)5;

or

::0000

00

:;

;

&=ca(in*

PSP yx

y

x

ss

y

x

s

s

y

x

=

=


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Inverse transformations

7% 891&22B

:

):

5)5

&=ca(in*

55

&


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12

12

12''

'

2

''

1'

MMMMP

PMM

P)(MMP

PMP

PMP

===

=

=

=

!ith

&Com+ine,ation---transformsecon,

---sformationfirst tran

Com+inin* transformations :

7% 89>&22B922


20/57

PT

PP

PTTP

PTP

PTP

''

'''

'

)5

:00:0

0:

:00:0

0:

:00:0

0:

)5)5

&Com+ine,

ontrans(atisecon,)5

s(ationfirst tran)5

2:2:

2:

2:

:

:

2

2

::22

22

::

yxxx

yy

xx

y

x

y

x

yxyx

yx

yx

tttt

tt

tt

t

t

t

t

tttt

tt

tt

++=

+

+

=

=

=

=

=

Com+inin* transformations 2

7% 89>&22


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6)56)56)5&sca(in*Composite

656565

&rotationsComposite

6)56)56)5

&onstrans(atiComposite

2:2:::22

2::2

2:2:::22

yyxxyxyx

yxxxyxyx

ssssssss

R

tttttttt

SSS

RR

TTT

=

+=

++=

Com+inin* transformations 1

7% 89>&22


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+ack-Trans(ate16ori*inHaroun,an*(eover


23/57

'''''

'''

'

)PT(P

)PR(P

)PT(P

R

yx

yx

,RR

R,R

==

=

16

26

:6

&pointaroun,an*(eover


24/57

)P)T()R(T(

)P)R(T(

)PT(P

)P)T(R()PR(P

)PT(P

'

'''''

'''

'

yxyx

yx

yx

yx

yx

R,R,RR

,RR

,RR

R,R

R,R

=

=

=

==

=

16

26

:6

&229210


25/57

PP

)P)T()R(T(P

'''

'''

+

=

=

:00

sin6cos:5cossin

sin6cos:5sincos

or

169:

xy

yx

yxyx

RR

RR

R,R,RR

R

: 2 17% 89>&229210


26/57

a*ain-+ackTrans(ate16ori*inH!-r-t-=cha(e26

ori*inH!ithcoinci,essuch thatTrans(ate:6

&point!-r-t-an,factors!ith=ca(e

F

Fxx ss

=ca(in* !-r-t- point :

F

: 2 17% 89>&210921:


27/57

'''''

'''

'

)PT(P

)PS(P

)PT(P

F

yx

yx

yx

,FF

ss

F,F

==

=

16

)26

:6

&point!-r-t-=cha(e

=ca(in* !-r-t-point 2

F

: 2 17% 89>&210921:


28/57

PP

)PT()ST(P

'''

'''

=

=

:00

6:50

6:50

or

6)5169:

yyy

xxx

yxyxyx

sFs

sFs

F,Fss,FF

=ca(in* !-r-t-point 1

F

: 2 17% 89>&210921:


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a*ain-+ack


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'''''

'''

'

)PR(P)PS(P

)PR(P

==

=

16)26

:6

&,irectionotherin=ca(e

2: ss

=ca(e in other ,irections 2

: 2 17% 89>&210921:


31/57

PP

)P)R()S(R(P

'''

'''

++

=

=

:00

0cossinsincos65

0sincos65sincos

or

)169:

22

2::2

:22

22

:

2:

ssss

ssss

ss

=ca(e in other ,irections 1

: 2 17% 89>&210921:


32/57

r,er of transformations :

#

0I

#I0II

#II

#

#

0II

#II

.atri mu(tip(ication ,oes not commute-

The or,er of transformations makes a ,ifference

Rotaton, translaton !ranslaton, rotaton

7% 89>&212


33/57

r,er of transformations 2

" Fre9mu(tip(ication&

P = Mn Mn9:M2 M: P

Transformation Mn in *(o+a( coor,inates

" Fost9mu(tip(ication&

P = M: M2Mn9: Mn P

Transformation Mn in (oca( coor,inates& thecoor,inate s#stem after app(ication of

M : M2Mn9:

7% 89>&212


34/57

r,er of transformations 1

penG& glRotate, glScale) etc-&

" Fost9mu(tip(ication of current

transformation matri" 4(!a#s transformation in (oca( coor,inates

" G(o+a( coor,inate version& rea, in reverse

or,er

7% 89>&212


35/57

r,er of transformations >

glTranslate();

glRotate();

# 0II

#II

#

Localtra"o

nterpretaton

Localtrans"ormatons&

# 0II

#II

Globaltrans"ormatons#

0I

#I

Globaltra"o

nterpretaton

7% 89>&212


36/57


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Direct construction of matri

x

y

$

AB

T

u

If #ou kno! the tar*et frame&

Construct matri ,irect(#-

Define shape in nice (oca(u,$coor,inates) use matri

transformation to put it

inx,yspace-

7% 89>&211921?


38/57

( )

::00:

or)

::

or);

=

=

++=

$u

!%&!%&

yx

$

u

y

x

$u

yyy

xxx

TBA

TBAP

Direct construction of matri

x

y

$

AB

T

u

If #ou kno! the tar*et frame&

Construct matri ,irect(#-

7% 89>&211921?


39/57

::00:

of);

=

=

$u

trrrtrrr

yx

yyyyx

xxyxx

MPP

&211921?0):KK):KK)an,

su+matri(orthonorma&

===

=

=

BABABAyy

xy

yx

xx

yyyx

xyxx

r

r

r

r

rr

rr


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ther 2D transformations

"


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PP'

=:000:0

00:


42/57

PRP'

PP'

6:B05as=ame

:00

0:0

00:

=

=

092>2


43/57

tan!ith

:00

0:0

0:

=

=

"

"

PP'

=hear

=hear they9as&

x=x+"y) y=y

or

x

y

7% 89>&2>292>1


44/57

Transformations coor,inates

Given 5x,y9coor,inates)

in, 5x,y9coor,inates-

692>B


45/57

Given 5x,y9coor,inates)

in, 5x,y9coor,inates-

Lamp(e& user points at

5x,y, !hats the position

in (oca( coor,inates$


x

y

x

y

5x0, y0

7% 89B&2>692>B


46/57

Given X& 5x,y9coor,inates)

in, X& 5x,y9coor,inates-

=tan,ar,&

X=MX 5o+ject trafo&

from (oca( to *(o+a(

ere&

X=M-1X 5from *(o+a( to (oca(


x

y

x

y

5x0, y0

7% 89B&2>692>B


47/57



ere&

X=M-1

X 5from *(o+a( to (oca(4pproach :&

9 Determine Mstan,ar, matriN M 5from (oca( to *(o+a( coor,inates an, invert


x

y

x

y

5x0, y0

7% 89B&2>692>B


48/57



ere&

X=M-1X 5from *(o+a( to (oca(4pproach 2&

9 construct transformation that maps (oca( frame to *(o+a( 5re$erse o" usual.


x

y

x

y

5x0, y0

7% 89B&2>692>B


49/57



ere&


4pproach 2&

:-Trans(ate 5x0, y0 to ori*inH

2-692>B


50/57



ere&


4pproach 2&

M-1= T5x() y(R5


x

y

x

y

5x0, y0

7% 89B&2>692>B


51/57

penG 2D transformations :

Interna((#&

" Coor,inates are four9e(ement ro! vectors

" Transformations are >> matrices

2D trafos& I*nore)9coor,inates) set) = 0-

7% 89&2>B92?1


52/57

penG 2D transformations 2

penG maintains t!o matrices& GL_PROJECTION

GL_MODELIE!

Transformations are app(ie, to the current matri) to

+e se(ecte, !ith&

glMatr"#Mo$e(GL_PROJECTION) or glMatr"#Mo$e(GL_MODELIE!)

7% 89&2>B92?1


53/57

penG 2D transformations 1

Initia(i3in* the matri to I& glLoa$I$ent"t%();


54/57

penG 2D transformations >

%asic transformation functions& *enerate matri an, post9

mu(tip(# this !ith current matri-

Trans(ate over Ot) t#) t3P&

glTranslate+(t#, t%, t-);

B92?1


55/57

penG 2D Transformations ?

penG maintainsstacks of transformation matrices-

T!o operations&

glPs.Matr"#()0.ake cop# of current matri an, put that on top of the stackH

glPo1Matr"#()0

9128


56/57

penG 2D Transformations 6

=tan,ar,&

glRotate(2, , 3, 2);

glScale(3, , 245);glTranslate(, 3, 6);

glt!"reC7e();

glTranslate(

, 3,6);

glScale(245, , 3);

glRotate(

2, , 3, 2);

Usin* the stack&

glPs.Matr"#();

glRotate(2, , 3, 2);glScale(3, , 245);

glTranslate(, 3, 6);

glt!"reC7e();

glPo1Matr"#();

Un,o transformation

=horter) more ro+ust

7% 9B&12>9128


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2D transformations summari3e,

9 Transformations& mo,e(in*) vie!in*) animationH

9 =evera( kin,s of transformationsH

9 omo*eneous coor,inatesH

9 Com+ine transformations usin* matri mu(tip(ication-

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2IV60!3!2D Transformations