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COMPUTER GRAPHICS AND

 ANIMATION

09 Transformations

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Transformations

• Some basic transformations

 ! Sca"in#

 ! Rotation

 ! Trans"ation

• Essentia" to man$ #ra%&ics a%%"ications'

E(am%"e cit$)%"annin# a%%"ication

 ! Trans"ation for %"acin# s$mbo"s *of trees+ b,i"-in#s+

etc. at a%%ro%riate %ositions

 ! Sca"in# for si/in# t&e s$mbo"s

 ! Rotation for orientin# t&e s$mbo"s'

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Transformations

• Matrices can be ,se- as o%erators to

%erform transformations'

•asica""$P’  1 P  ' T 

P  1 ori#ina" %oint

T  1 transformation matri(P’  1 res,"t %oint

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Sca"in#

• Sca"in# base- on*0+0.

• P  is sca"e- b$ s x  

a"on# t&e ( a(is an-b$ sy  a"on# t&e $a(is'

• T&,s x’ = s x  . X 

y’ = sy  . y 

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Sca"in#

• In matri( form

 x’ = s x  . x + 0 . y 

$2 1 0 ' x  3 s$ ' y T&,s

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Rotation

• Center of rotation at

*0+0.'

•P  is rotate-

ϑ

-e#rees co,nter)

c"oc45ise'

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Rotation

• In matri( form

 x’ = x cos – y sinϑ ϑ

y’ = x sin + y cosϑ ϑ• T&,s

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S&earin#

• I2m "ost at 5or-s+ so

-escribe t&is $o,rse"f'

•dx = dy / m  = y / m

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S&earin#

• In matri( form

 x = x + y/m

y = y 

• T&,s

5&ere a = 1/m

• 6&at abo,t s&earin# a"on# t&e $ a(is7

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Trans"ation

• Mo8e P -( ,nits

%ara""e" to t&e (

a(is an- -$ ,ntis

%ara""e" to t&e $

a(is

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Trans"ation

• 6e #et

 x’ = x + dx 

y’ = y + dy P’ = P + T 

• Ho5 5o,"- t&e transformation matri( be7

P’ = P . T T  1 777

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Transformation

• 6e 5ant to ca%t,re a"" t&e tranformations

in t&e form of a matri('

• Ma4e it so t&at P’ = P . T  can be a%%"ie-

for a"" 4in-s of transformations+ including

trans"ation'

• Is it %ossib"e to &a8e a transformation

matri( incor%oratin# trans"ation7

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Homo#ene,s Coor-inates

• Sca"e+ rotate+ s&ear P’ = P . T 

• Trans"ation P’ = P + T 

• Transform P*(+$. into a &omo#eneo,s

coor-inate P&*(+$+5.

• T&is coor-inate s$stem &as m,"ti%"e8a",es for same %oints

 ! E'#' *+:+;. an- *<+;+=. are t&e same %oint'

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Homo#eneo,s Coor-inates

•  At P+ 5 1 =

• P*(>5+ $>5+ =.

• P= an- Pre%resent t&e same

%oint'

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Homo#eneo,s Coor-inates

• ?et P*(+$+=. an- P2*(2+ $2+ 5.

 x’ = ax + by + m

y’ = cy + dy + n

• T&e transformation matri( c&an#es from( to :(:+ t&,s

• P2 *(2+ $2+ =.

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Transformation O%erators

Sca"in# S&earin#

  S(s x , sy  ) Shy (a) Sh x (b)

Rotation Trans"ation

  R( θ  ) T(dx, dy)

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Transformin# ?ines

• To transform a "ine+ $o, on"$ nee- to

transform t&e en- %oints' T&e ot&er %oints

5i"" @fo""o5'

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Com%osition of Transformations

• T&e %rocess of a%%"$in# se8era"transformations in s,ccession to form oneo8era"" transformation

• If 5e transform a %oint P  5it& M= *res,"tin#in P’ .+ an- t&en transform t&e res,"t ,sin#M 2  *res,"tin# in P’’ .+ 5e #et

P’   1 P . M1P’’ = P’ . M2 

T&,s P’’ = P . M1 . M2 

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Com%osition of Transformations

• If 5e %erform n s,ccesi8e transformations

on a %oint P + 5e #et

P’ = P . M 1 . M 2  . M  . ... . M n

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E(am%"e

• Ho5 -o 5e rotate a s&a%e -e#rees fromϑ

an arbitrar$ %oint P7

• Bor eac& %oint in t&e s&a%e Pi+ 5e %erform

t&ese : ste%s

 ! Trans"ate s,c& t&at P is t&e ori#in *M=.

 ! Rotate *M9.

 ! Trans"ate bac4 to ori#ina" %oint *M:.

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E(am%"e

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E(am%"e

• ?et P*(+$.

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E(am%"e

T 1 M= ' M ' M:

  1

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Data Str,ct,re

• Type TMatrix = array [0 .. 2, 0

.. 2] of real

• Type TPoint

< x : real, y : real>

• Type THC = array [0 .. 2] of

real {hoo!eno"# $oor%inate&• Type THC

 < x : real, y : real, ' : real>

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Re8ie5 ,estions

• 6&$ -o 5e ,se :(: matrices instea- of (matrices for D transformations7

• Deri8e t&e transformation matri( for sca"in# a %ointfrom *0+0.'

• Deri8e t&e transformation matri( for rotatin# a%oint from *0+0.'

• Deri8e t&e transformation matri( for s&earin# a%oint on t&e ( a(is an- $ a(is'

• Deri8e t&e transformation matri( for trans"atin# a%oint'• In #enera"+ &o5 -o 5e %erform a sca"in# or rotation

o%eration from an arbitrar$ %oint7

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E(ercise

• Bin- t&e transformation matrices for t&e

fo""o5in# o%erations

 ! Trans"ate a %oint b$ *<+ .

 ! Sca"e t&e ()com%onent of a %oint b$ 0' an- t&e$)com%onent b$ : from *0+0.'

 ! Rotate a %oint < -e#rees c"oc45ise from *0+0.

 ! S&ear a %oint on t&e "ine $ 1 ( a"on# t&e ( a(is'• Perform t&e abo8e transformations on t&e

%oint *:+<.

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E(ercise

• Define t&e transformation matri( for t&e

fo""o5in# o%erations

 ! ertica" f"i% *ref"ect to ()a(is.

 ! Trans%ose %oint *P2*$+(..

 ! Ref"ection from an arbitrar$ %oint P'

•  An obFect is to be ref"ecte- t5ice as far from

t&e "ine ( 1 ' Bor e(am%"e+ if P is *0+0. an- Pis t&e res,"t of t&e transformation+ P is *=+0.'

Deri8e t&e transformation matri('

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E(ercise

• Pro8e t&at R *θ 1. R *θ 2 . 1 R *θ 1 + θ 2 .' 6&at

-oes t&is mean7

• Pro8e t&at T*(=+ $=. ' T*(9+ $9. 1 T*(=3(9+

$=3$9.' 6&at -oes t&is mean7

• Pro8e t&at S*(=+ $=. ' S*(9+ $9. 1 S*(=(9+

$=$9.

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E(ercise

 An ort&o#ona" matri( is a matri( 5&ic&+

5&en m,"ti%"ie- b$ its trans%ose+ res,"ts in

t&e i-entit$ matri(' Hence+ t&e in8erse of an

ort&o#ona" matri( is t&e trans%ose of t&ematri('

•Pro8e t&at rotation matrices are ort&o#ona"'

•6&at -oes t&is mean7

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E(ercise

Bin- t&e transformation matri( for t&efo""o5in# o%erations

•Sca"in# an obFect from t&e ori#in %oint b$

in ( an- $+ fo""o5e- b$ a trans"ation b$ *=+=.'•Trans"ation of an obFect b$ *=+=. fo""o5e- b$a sca"in# from t&e ori#in b$ in ( an- $'

•6&$ are t&e t5o res,"tin# matrices-ifferent7

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E(ercise

• 6&at D transformations are -one to

transform t&e s&a%e on t&e "eft to t&e

s&a%e on t&e ri#&t7 E(%"ain' Deri8e t&e

transformation matri('

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E(ercise

• 6&at D transformations are -one to

transform t&e s&a%e on t&e "eft to t&e

s&a%e on t&e ri#&t7 E(%"ain' Deri8e t&e

transformation matri('

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E(ercise

T&e fo""o5in# matri( M is a transformation matri(

?et M= an- M be transformation matrices s,c& t&atM 1 M= M'

Define M= an- M if•M= is a trans"ation o%erator an- M is a sca"in#o%erator'

•M= is a sca"in# o%erator an- M is a trans"iationo%erator'

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E(ercise

•  A %oint is to be transforme- in s,c& a 5a$ t&at

t&e -istance from t&e %oint *:+ <. is &a"8e-' Bor

e(am%"e+ if P is *=+ . *-istance 1 √ from *:+

<.. an- P is t&e res,"t of t&e transformation+ P is*+ :. *-istance 1 √ from *:+ <..' Deri8e t&e

transformation matri('

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E(ercise

e"o5 is a rectan#"e s%ecifie- b$ its fo,r 8ertices A+ + C+an- D' Determine a set of in-i8i-,a" D transformationmatrices t&at transform t&e rectan#"e ACD intorectan#"e A22C2D2 as s&o5n be"o5' Note t&at t&etransformations s&o,"-+ in t&e en-+ transform 8erte( Ainto A2+ into 2+ etc'