3d Viewing Trans

7/29/2019 3d Viewing Trans

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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


3/51

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

-


4/51

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


7/51

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)

+ , - - -


8/51

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


9/51

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


10/51

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


14/51

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


15/51

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


16/51

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


19/51

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


21/51

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


22/51

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

()(


23/51

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


25/51

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)


26/51

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


27/51

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


28/51

w here af t er cl ipp i ng , use

MCVV3DVP =

.)Z,Y,T(X vminvminvmin

ZZ

YYXX

Svminvmaxvminvmax 22

, ,.


29/51

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 .


30/51

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


31/51

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


32/51

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


33/51

he Com put erraphic s Pipe l ine


34/51

The Cam era Model


35/51

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


36/51

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


37/51

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


38/51

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


39/51

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:


40/51

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 )


41/51

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:


42/51

ab o ve t w o eq u a t i o n s:

.

nTh e n

:

.b


43/51

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


44/51

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


45/51

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 )


46/51

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


47/51

nf

cot

'

nf

0nf

.00

'2nfnf

..

n

nfnf

nn

110


48/51

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


50/51

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


51/51

End of Lectures on

Pro ection Transformationsand

ew ng pe ne

3d Viewing Trans

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