3d viewing trans
TRANSCRIPT
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Projection Transformations
and
ew ng pe ne
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App ly3 - D w o r ldcoo rd ina te
Cl ip ag ain stcanon ica l
n o r m a z n gt r a n s f o r m a t i o n
o u t p u tr i m i t i v e s
ViewV o l u m e
Pro jec tTr a n sf o r m i n t o
v iew or t in
Pro jec t ion2D d ev icecoo rd ina tes
2D d ev icecoo rd ina tes
fo r d i sp lay
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Canon ica l v i ew vo lum e fo r ara l le l p ro jec t i on i s de f in ed by s ix p lanes:
X = 1 ; Y = 1 ; Z = - 1 .
X = - 1 ; Y = - 1 ; Z = 0 ;
X o r Y
1
BP
- Z- 1
-
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Canon ica l v i ew vo lum e fo r er s ect iv e p ro jec t i on i s de f in ed by s ix p lanes:
X = - Z; Y = Z; Z = - 1 .
= ; = - ; = - m in ;
X o r Y
1FP
- Z- 1
- 1
BP
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VP, VRP, VUP, VPN, PRP, DOP, CW , VRC
v
(umax, vmax)
VRPCW
VPN(umin, vmin)
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v
u v
VRPCW
u
VPN
(umin, vmin)
v
n
VRPCWVP u
VPN
ar b i t r a r y 3 D v i ew
n COP/ PRP
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Spec i fy ing an Arb i t r a r y 3D View
V i e w i n g Example Valuese e e
VRP WC 0 , 0 , 54 16 , 0 , 54 0 , 0 , 0
VPN ( W C) ( 0 , 0 , 1 ) ( 0 , 1 , 0 ) ( 0 , 0 , 1 )
VUP ( W C) ( 0 , 1 , 0 ) ( - 1 , 0 , 0 ) ( 0 , 1 , 0 )
, , , , , ,
W i n d o w ( - 1 , 1 7 , ( - 1 , 2 5 , ( - 5 0 , 5 0 ,
-1 , 17 -5 , 21 -5 0 , 50Pro jec t ion
Type
F & B(VRC)
+ , - - -
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St eps for im plem ent ing nor m al izingt ransfo rm at ion mat r ix fo r
ar al le l ro ec t ion
Tr ans lat e t h e VRP t o o r ig in
Rot a t e VRC such t h a t VPN ( n -ax is) a l igns- , -
su ch t ha t DOP is pa r a l lel t o t he Z-ax is
Tr anslat e and sca le i n t o pa ra l le l - p r o j ect i onVV
VRP)T(RSHTSN arararar
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St ep 2 i n n o r m al izi n g t r a n sf o r m at i on s:
Ro t a t e VRC such t h a t VPN ( n -ax is) a l ign sw i t h Z- ax i s a l so , u w i t h X an d v w i t h Y
v
VUP
max, maxVP
VRP
u
VPNm n, m n
n
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Ex pr essions fo r St ep 2 m us t be de r i v ed .
I m p l em en t u si n g t h e co n cep t o f co m b i n edt r an sf o r m at i o n ( r o t a t i on ) .
Tak e R =
0)sin()cos(0
1000
0)cos()sin(0
Ro w s ar e u n i t ve ct o r s, w h e n r o t a t ed b y Rx ,
w a g n w e an ax s r esp ec v e y .
ar e r o t a t ed b y Rx , t h ey f o r m t h e co lu m n
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0001Rx =
0)sin()cos(0
1000
Row Vectors: [ 1 0 0 ]
Z
[ 0 sin() cos() ]Column Vectors {con side r Rx ( -) ,
n t s case :[ 1 0 0 ]T
cos -s n [ 0 sin() cos() ]T
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Con s ider a gen era lYscenar io o f com b in ed
r o t at i on s an d u se t h eP3p r oper t y de r i ved basedo n t h e o r t h o g on a l i t y
P2
P
o e m a r x .Y
Pn
Z
3Befo reT rans fo rma t ion
XP1 Pn
P
Af te r
Z
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Let t h e e f fec t i ve r o t a t i on m at r ix b e
a com b ina t i on
xxx
rrr Yo f t h r ee r o w s as: rrr 3 P2Pw h e r e , X
T 1 2
z 1z 2z 3z
P PR = r r r =
P P
n
T 1 2 1 3P P X P P= =x x x x1 2 1 3
P P X P P3
Pn
T
= =
XP1
y 1y 2y 3y z x Z 2
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Y Y
3
P2P P3
X X
n
n
Z P2
Th u s t h e r o t a t i o n m at r i x o f st ep 2 i n
,f o r m u lat ed a s:
0rrr 3x2x1x
0rrr 3y2y1y
0rrr 3z2z1z
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St ep 2 i n n o r m al izi n g t r a n sf o r m at i on s:
Ro t a t e VRC such t h a t VPN ( n -ax is) a l ign sw i t h Z- ax i s a l so , u w i t h X an d v w i t h Y
v
VUP
max, maxVP
VRP
u
VPNm n, m n
n
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z
;VPN
R w h e r e
RP x ;RVUP
RRRand
f o r pa ra l l e l p ro j ect i on ( W CSVV - > PPCVV) , i s:
VRP)T(RSHTSNparparparpar
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Th e ove r a l l co m b in ed t r a n sf o r m at i on m at r i xo r p ar a e p r o ect on - > , s:
parparparparw h e r e ,
001 parshx ;xpar opshx
0100
par
parSHz
do 1000 parz
s ydop
DOP DOP
-z
VPN VPNSide v iew o f
-z
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Th e ove r a l l co m b in ed t r a n sf o r m at i on m at r i xo r p ar a e p r o ect on - > , s:
parparparpar
;minmaxminmax FP
vvuuTpar
122minmaxminmax BPPvvuu
par
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transformation matrixfor perspective projection
v
VRPVP
uVPN
n
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Canon ica l v i ew vo lum e fo r perspec t i ve p ro jec t i on i s de f in ed by s ix p lanes:
X = - Z; Y = Z; Z = - 1 .
= ; = - ; = - m in ;
X o r Y
1FP
- Z- 1
- 1
BP
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Ste s for im lementin normalizin
transformation matrix for
r an s a e e o o r g n
V V -w i t h Z- ax i s ( a lso , u w i t h X- an d v w i t h Y- ax i s)
r an s a e su c a o r s a eo r i g in
Sh e ar su ch t h a t cen t er l in e o f v i ew vo lu m e( VVCL) becom es z -ax is
Sca le su ch t h a t VV becom es t h e canon ica l
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Scenar io o f t h e cr oss-sect ion o f t he VVa t er r st t r ee t r an s or m at on s.
X o r Y
CW
- Z
VPN
VRPTRPRPTSHSNarerer
()(
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Com ar ison t h e ov er a l l com b in edt r a n sf o r m at i on m at r i ces f or :
PARALLEL PROJECTI ON:
VRP)T(RSHTSN parparparpar PERSPECTI VE PROJECTI ON:
VRP)T(RPRP)T(SHSN parperper
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App ly3 - D w o r ldcoo rd ina te
Cl ip ag ain stcanon ica l
n o r m a z n gt r a n s f o r m a t i o n
o u t p u tr i m i t i v e s
ViewV o l u m e
Pro jec tTr a n sf o r m i n t o
v iew or t in
Pro jec t ion2D d ev icecoo rd ina tes2D d ev icecoo rd ina tes
fo r d i sp lay
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dd
dd
z
p
z
en er a ze
f o r m u l a
dz
d10
z
y
p
z
yo p er sp ec v ep r o j ect io n m at r i x :
Zd
Zd
Z00 ppp
gen
1
Z1
00
p
zzPP
P(xp, yp, Zp)L
P(X,Y,Z)(dx, dy, dz)
(0, 0, Zp)
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Coor dinat e S st em s and Mat r ic es
Per s ect iv ePara l le l
3 - D m o d el in g Mode l ing
coo rd ina tes
ViewOr ien ta t i on
o rCoord ina tesR.T(-VRP)
- m a r x.
Cont
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View ViewMa in
ew
re fe rence
m a t r i x m a t r i xs
Cl ip , t r ans fo r m
in t o 2D scr een
or m a zep r o je c t i o n
coo rd ina tes
MCVV3DVP
ev cecoord ina tes
M . Sp er . SH p ar . T( - PRP)
S . T . SH
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w here af t er cl ipp i ng , use
MCVV3DVP =
.)Z,Y,T(X vminvminvmin
ZZ
YYXX
Svminvmaxvminvmax 22
, ,.
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The 3D View ing Pipe l ine
Ob j ect s a re m odeled i n ob j ect ( m odel i ng )
space.
Tr ansfo r m a t ions ar e app l ied t o t he ob j ect s
t o p o si t i o n t h e m i n w o r l d sp ace.
View pa ram et ers ar e speci f i ed t o de f i nee v ew v o u m e o e w or , a p r o ec on
p lane , and t he v iew po r t on t he scr een .
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ec s ar e c p p e o s ew v o u m e.
e r esu s ar e p r o ec e on o ep r o j e ct i o n p lan e ( w in d o w ) an d.
en o ec s ar e en r em ov e .
Th e ob j ect s a re scan con ver t ed and
t hen shaded i f n ecessary .
Fl h t f t h 3D Vi i Pi l i
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Flow c har t of t he 3D V iew in Pi e l ine
Objec t
Ob jec t
, ,T rans la te
W o r l dSpaceW o r l d
S aceSpeci f y View ,l l N r m l
App ly No rm a li zi ng
I l l u m i n a t i o n ,Backface
Clip
Space
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Ey e I m a g e
Trans fo rma t ionRemoveHidden
Space Space
u r aces
Shade,
ap oV i e w p o r t /
Dev ice
screenoor n at es
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he Com put erraphic s Pipe l ine
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The Cam era Model
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The Cam era Model
W e speci f y ou r i n i t i a l cam er a m odel by.
1 . A scen e con sist in o f o l on l e lem en t s
each r ep resen t ed by t he i r v e r t i ces;
. p o n a r ep r esen s e cam er ap o si t i on : C = [ Cx , Cy , Cz] ;
3 . A p o in t t h a t r e p r esen t s t h e cen t er - o fa t t en t i on o f t h e cam er a ( i .e . w h er e t h e
cam er a is l oo k i n g) : A = [ A x , A y , A z] ;. e - o - v ew an g e, ,r ep r esen n g e
an g le su b t en d e d at t h e ap ex o f t h e v i ew in g
p y .
The speci f i ca t ion o f nea r
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The speci f i ca t ion o f nea ran ar ou n n g p an es.
Th ese p l anes con s ider edp er p en cu ar o e
d i r e ct i on - o f - v i ew ve ct o r
n an d f f r o m t h e,
respec t i ve ly .FarPla
eC
Th e View i n P r am id
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Th e View i n P r am id
v iew ing space
Far
vPlan
Th e im age spacev o l u m e :
1wv,u,1
Side v iew o f t he v iew ing space
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Side v iew o f t he v iew ing space
Fa
rPla
earPlane
ne
f
Der ivat ion of t h e v iew in t r ansfor m at i on
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Der ivat ion of t h e v iew in t r ansfor m at i onm at r ix , i n t e rm s of c amera pa ram ete rs :
d.u d.v d.u d.v d.w(u,v,w) ( , , d) ( , , )
w w w w wThus
(u, v, w, 1) (d.u, d.v, d.w, w)
P( u , v , w )u o r v
u v w
PP
O ( COP) - w
Express as transformation:
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Express as transformation:
001000d0
d1001d00 d
000d
wd.wd.vd.u00d0
1wvu
0000
Tr a n sf o r m at i on o f t h e f in i t e ( t r u n cat ed )
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Tr a n sf o r m at i on o f t h e f in i t e ( t r u n cat ed )v iew in g p y r am id t o t h e
cu be ( CVV) , - 1 < u , v , w < 1 .Th e im age space
Let us f i r s t ana l yze w -ax i s on l y . 1wv,u,1 Use t e t r an s o r m at on m at r x :
su ch t h a t ,
;P
, , - , ,
an d- -
, , , ,
So l ve f o r a ram et er s a and b u sin t h eabove equ a t ion s:
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ab o ve t w o eq u a t i o n s:
.
nTh e n
:
.b
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Hence t he 0001t r a n sf o r m at i on i s:
0010
nfP
0
2f.n00
a a ou u an v - ax s r an s o r m a on si n t h e py r a m i d ?
u an d v - a x i s t r a n sf o r m at i on s
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i n t h e p y r am id
u o r v
n . tan
/ 2.
O COP - w 0 , 0 , - n 0 , 0 , - f
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Tr a n sf o r m at i on s f o r t h e t w o p o i n t s
a re as fo l l ow s:
( 0 , n . t an ( / 2 ) , - n ) ( 0 , f . t an ( / 2 ) , - f )
/ 2O ( COP) - w ( 0 , 0 , - n ) ( 0 , 0 , - f )
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0010
Desi r ed no r m a l i zed 3 - D
co o r d in at es f o r b o t h t h e
1
nf
n00P
po in t s : [ 0 , 1 , + / - 1 , 1 ] .
0nf
.n00
.1f/2)f.tan(0 P
f2nfnf
f/2)f.tan(0
ff/2)f.tan(0
Th u s m o d i f y Pt o b e:
000)2/cot(
t
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nf
cot
'
nf
0nf
.00
'2nfnf
..
n
nfnf
nn
110
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I t s in v e r se h as t h e f or m :
000tan )2/(
00tan0 )2/(
2fn
000P 1
nf
100n
The V iew ing Trans for m at ion Mat r i x
.df
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000)2/(d.cot
.df
2n)d(f 00)2/(d.cot0 n
0000 nf
0010or
00d0
000d
P
00cot(0
000)cot(
)2/
2/
0000 d
100
0000
1d00
1
nf
n00P
00d0
000d
0nf
.00
...
0000
1d00
or
us ing t he r egu la r ex p r ession o f Pd
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00)2/(cot0
1
2fn-n)d(f00
0000
0010or
00d0
000d
P
00cot(0
000)cot(
)2/
2/
0000 d
100
0000
1d00
1
nf
n00P
00d0
000d
0nf
.00
...
0000
1d00
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End of Lectures on
Pro ection Transformationsand
ew ng pe ne