Cate
UNTT-9
importat foadneA. any qonapbic actae.
It s change the obiect podi er , Sfze and
Ogfentadion by 8ing tsanslation Bcaiy
Beaing and
pnocess 1eapectiely
Toiansdatoo
Rotatan
(1)
Scaling and athe
Sheaing
’ Each single antdy
Tt can be denated by a urique.
Tynanslatiam the.
the po8i fon of ary ohe ct n Xdirectae
chamge
by amount and y-diaectoo bu ty amound
Tt s he stralgh Jne mauement ot
pos1ion cto anathen 185. called
Tnanlaton.
pesttaned trom
Heas the abject S _posidned fr
Jocatian_to anothe
tsanlaian can lso be intorgKe.ted o the addttn
can stant vecitor to ereny
hiftrng he cstgin patnt 6.
of he coor dinote sysiem
Toanslaton af psint
coosdine poston Cx) to anothen Cx, we add
dgebaajcally the tanslao dstances The Tr and
Ty to aTiginal coondinate
The toanslaton palT Ty) s called aA shift
vectas
pay) a foint befose toangdatin
(
Point atte tanslattn
Tnansdation panamtens dy
Cuhene To-ndatn în direction by tr
tc the
-dinecticn ) be r'=c+t
and toan latnIn
in -d ectn by ty the ne prsiton
Tn 8-drecton wt be 9'-y+ty
So zanstoon function fö
Mataix nepneientatin
y' dy
Hornogen eeus-noto
(hefore trans Jation) Caften tanslatom)
1) ScaDdIng t Scaling a geometac toanstormadon
to aten the Size of an obËect ([thesn ncasasu
o decnease) by a Scaltng tactos
directin and S, in y- dinectham
’
sed to alten change the slze af chËecta
The change s done sing scaling factor
The.ne ane tuo Scabng factor /e. Sr in
dnecttonSy Y-dtnecom.
I the oaigina
postion iss a and y. Scalin
ane and sy then the aue of
Caax dinat& aften &oaling unll be o and
Dsta
pose patnt P ) ss given then the soaled
giuen by.
Sy
rlatatx orm
S
Sy
Homogeniaus (ax3) sealimg -matix
S 8o
&y
p' = S.ee Scale, functian
thete
S=
O- Sy
(betine scahng Catten Scaling)
Rotadon It S K_a panCEAR of changing the
angle the dbject
clockatse 4 antclackuräe
Rotaion CAn be
Date
cJockunse. directoo.
R SinA
sine CoSe
Matax fos Rotation an antclo ckuise drrt
Sine
R
|Sine
Matatx foa Hanagensaua co-ordinate s natasion (clackuis
R -Sine
Matsxfos HamogeneLA co-rdinatei notaanantelackus
R Sine
(lee
uppiep an nihal psnt hich smake
angle uith
HomogetoUa Cerodintea
HmoqenecH CooLdnat stng rmean expelny ea ch condnd
tcant mation colation matat mutpk cato
Toandatn.
y' Jy
1
&) Rotatsn
Co
y y
Scaling
y' Sy y
Date
SuccEe and CaTnpoilt aTD Toonsfosmaticn
fucceASle toansfasnatin
anithea.
heae che apphin smplfmatinio a sealence whege
tianitacnat?rn beceoe)the tnput
the next
Each tranatemadten S apled Sepaatcy
Caoxdinates.
mlakt fst tanslate. object then otte tt
and fnally
Campste tansfnaton t Repesent the Cmknatim of
thege toanyy mation az matax qpeaatho
The gaesuting Cacxdipate
Cangare aoht can hen he apl
be
tn achlee_the Same gcau
ha
the obsects
dhe Genllenx af taanstmoteni
into ne Bingle
ntmaon tht S cOUalert the toanatnho tht
peatosmed ne ofteg one
don fosnaton T ond dhon we asply tantl' T:
Jhe voy cauitralet feot
Can cbtain hy autt ply'ng I4Te with cah ott
of the mage to
the coordintes.
taansfamed final mag
Psnoblem. Constden hae
facdr
and Te (atatlon transmaton nn.
then te aply Ta
Ta (eflectin txanstomain about arig!n)
Ans The
De,4)
eo,o)
Dare
e pcafsn Scaltng toanstaairo
Rpresendaton
y' O Sy
Cooxdioate olo.o)
0-5
Cocadinate
y
L41
y'
fs Candinate
s
o-54
Date.
QD object afteo scaling
ca2)
B,o)
The Cocioo of Rotati tsantmatto of
Cos S]n0
-Sn0 Co0.
y
Sin 9 1.
-!
fox canxdinat cle):
e 2
-!
y'
QD obecd aaten, otaty abat osfgin by 9o' angle
loo.). D(e,0)
B(o. c(a,-2)
qaten fg-3by
abot cála n
The Candita chjcct about
Cemdtnate oloo):
-!
-!
Ceodtnat Blo.-o]
foa Conndinat clô.-2)
y'
Candlnate DR,0)
The final QD ob<cot aftes nefectng abat adtgn
we get
c-R2)
2
o,0)
ctt-e
Q9 Viewlna taanafosmasten Lged t toanate te urcld
Co-CTdinatei to deMce coc.asdinates Cnanitox)
Tuo -dnen Sinal
arox ld abject trto positicn potnt Lwhich s nedave to the
ieLliny waluene espechlythe points bebtnd the teute
feuing
Medeling
CACKdinate kan&trsnatir
Naana kzed Device Tndepcrsert
clping ranstomadten Foangfonatior
VteLspaat tntmatiom fhysical
Taansforoation Co- asdinson
Scan Comyensfn
Devce
Tnage Cooxchnates
WRndaw The method of sele cting and enlaging a postn
ot a cnAbwhg s caled lndowng. The aea cheter
fos this display A caled ndo
’ he wndaw is Selected byCoCongdinate -ansa
Soae nes intenested in
Hhe obËe ct and not n ftull chËect So.
Date
trm sginay bx. Tals bo ual éndose deatxed o
oteested
aneaof the ctjeat Such an imaginasy
imaginasy hax is
alel a windaua
tndao The cligpng wÙnda select ahat we
dizplay
Vi eLapoTt uhndoa t The ewpost otnda tndicate whene
Stobe ie on the atput clece
Tnotheuoxd e Can say that vMesprt S an
ndaplay devtce to hich a aindaw s Ynaped
AJmost al) Q0 and 30 gnphica ckagel pmvide
of difintng Mewpaxt Sze an the sSOneen.
Aosaible to detomine many eapods on difeset
I} 8 Same object sn a
ot display and ves the
1n each eprt
diffenent angle
Cacxdn ate
deine the diagasan
Yumin uma
Xumin
Destce Cooxdinate - I s he Scean caondinaded
uheae the cbject y to be dhsplayed hke
XuminXumax Numin umac
Calcukdtmm of ehndow do iepart
It may be prs srble that the Sze of the Meaprst 1s
ch Snalle
Smallen greate han he wndaw
Tn thee caes ,we hae to in Cneoaie
the síze of the window aclasding do the. ieLpent and
for thiS ure need &ome mathemahcal calcuWaion&.
Apnt
Cxresponding paint
hae o caleulate the pant Cu,tu)
Ngsmalized Pant on windowXo-Xuotn yo-umin
Xomar-Xumn Ywmax - tomin
Nosma)izad Psrt an Mhewpast Xy-ymin
Xvmag-Xvmin
Na the sehthve pot peslfianm of the cie ct tn wndau
qnd Cne Same
XCasndinate
Xw- wmin Xy - XVvnjn
Xmase- hwmin Xvnax - Xvmin
Conndinates
Swma-Yw min Yymar Yumin
" So, aften caleulattna tx X andy caordinates weqet
Xy_s Xy oin +(Xu- Xumn)
Sr
(aumin) Sy
(ohereS S Scahy tactor af caasdnote and
Sy ot caTydnate
S Xvnas-Xvmin
Xwmax- Xwmin
Yumax - Yynin
Date
39 Tyanafos machons Matstx
3D Page
Tianstnmaton
and chargtng the pasitcn of an extshng wnyohect
of madhhyng
Computa gaphta
D A5an&fasnatlen,
manipulates the new of 3D abject
Sds csigigal postham by snaly edfytrg the phya1cal
attabute f that chiec medhods of
tsansomatlon Itke toanlaton gotn
canshean etoTC
nes YaJntain these sate
Lacattn of oeats
sbËecs ahh xpect o othes Can cAsiy
fasallelsa malntain.
als. -prekoved
analam Bestoc
the pastons
colect the mae
Cooxdinctes.
Tt
changng
ts hy
the toape(t wth cbject BD a of
elatye thechags) Tianslatn Tanlt
in
'
trmaticns
- Trang 7ypes
of
ct
oject biape the changlng
object, in
toanatomans
ane 30 3:
aDthan Camplex toangtfosmaTions
pane 3.D dnw be Can
Featunes
Cat
ae
3D bucd
rale fn30 eMsonmert
sT Rotatn deals
s theetion: Hee obiect can aotate
chect panle) doa spectc axc s9 hat the
Cactdinate ushene the obect aotadca malnt
unchanged and the emainy dun get changea Three
khnds of Such aotiens psstbe
Rotaicn abrut the Xaxt
Rotatten abeut the yaxs
Rotdn
Rotatem aboutthe 2-axs.
Rotat'n the X-axls - Tn tthts kind af sotaton
about
the cbtect tated pae,culleJ to the X- axts
uhene he x coosdinate. emalns unchangtd and the
of cooxdinatea y and Z cndy change
Cunat des potnt n1hacasdnate PXy.z)
S nade. to aotate pamalleJ tothe
in 3D pace
pancipa axds (x-ada) The Caxdt nate paát7m uld
to p a pa9z)
change
matax s wed to calculte
A gototen anstrathon
the nelo CooTdinate
note Comdi the
change
X-Cocsdinate
ongle utatin shene
CosO -Sino
Sh0
Dale
Date
Rotatem abet the yaxls In tts kind of£ gtatiD
the cyect gutated pasnae
the y Y-axts
of the
Ceosdgnate
te0 mernalns unchanged and the
cODSdinates X and ~ any change
-Stno
Csnsldesna psnt
30 space made to otte pasalle) th the.
psiocipal ants (-axis ). The eacdtnte prsltan wmuld change
p(x9,z)
changs
ZnZoCasa- T Sn
