3D Viewing. Vanishing Points. Two ways Intersection of transformed lines Transformation of points at infinity. Y VP z. VP x

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2 Vaishig Poits Two ways Itsctio of tasfomd lis Tasfomatio of oits at ifiity Y Y VP z X VP x X Z

3 Pla Gomtic Pojctios Paalll Pscti Othogahic Axoomtic Obliq Sigl Poit Timtic Dimtic Isomtic Caali Cabit Two Poit Th Poit

4 Imlmtatio Isss Mo fom Itfac oit of iw Y V U Ey N X Z Wold Coodiat Systm (WCS) Viwig Coodiat Systm (VCS)

5 Viw Coodiat Systm (VCS) Viwig coodiat systm Positio ad oitatio of th iw la Extt of th iw la (widow) Positio of th y Viw Pla Viw Rfc Poit (VRP): th oigi of VCS scifid as ( x, y, z ) i WCS: ct of th sc Nomal to th iw la ( x, y, z )

6 Viw Coodiat Systm (VCS) Viw Pla Nomal Dictio (Viw Pla Nomal VPN) ( x, y, z ) Us may oid omalizd cto.g. x si φ cos θ y si φ si θ z cos φ X Z φ θ Y

7 Viw Coodiat Systm (VCS) Viw Pla Dictio is a it cto ititily cosodig to cto cto is scifid by th s i WCS (.) / Dictio x ( Lft Hadd)

8 Widow ad Ey Viw Coodiat Systm (VCS) Widow : lft, ight, bottom,to (w l,w,w b,w t ) gally is ctd at VRP (oigi) w w t Ey : (,, ) Tyically (,,-E) w l w b

9 Tasfomatio fom WCS to VCS Y O (x, y) ( x y ) ( a ( a b) + b) M + O X

10 Tasfomatio fom WCS to VCS Poit objct is std as (a,b,c) i VCS (x,y,z) i WCS z y x z y x z y x M

11 [ ] [ ] [ ] T M M c b a M c b a z y x ) ( ) ( + Tasfomatio fom WCS to VCS Cosio fom o coodiat systm to aoth Thfo a(-)., b(-)., c(-).

12 Tasfomatio fom WCS to VCS I Homogos Coodiats (a,b,c,) (x,y,z,) A w A w? M? T?

13 [ ] taslatio T T T M M M c b a ) ( I Homogos Coodiats -M T (-.,-.,-.) ( x, y, z ) xyz A w ' ' ' ' ' ' z y x z z z y y y x x x z y x T w M A Tasfomatio fom WCS to VCS

14 Tasfomatio fom VCS to Viw Pla (,, ) t *(*,*) * t tt Paamtically (t) (-t)+.t

15 Tasfomatio fom VCS to Viw Pla O - la, (t) t t t + * * ' ' ' ) ( ) (

16 M Wh y is o -axis * /( - ), * /( - ) Matix fom (*) Pscti Tasfomatio Tasfomatio fom VCS to Viw Pla

17 Usig Pscti Tasfomatio M Tasfomatio fom VCS to Viw Pla (sdo dth),),, ( ),, ( * * * * * * * M

18 s M *(,,,)M s M q : i WCS mas to *qa w M s M Tasfomatio fom VCS to Viw Pla If y is off -axis w ha aoth matix

19 Viw Volm Ey Viw Pla, Fot Pla F Back Pla B

20 Viw Volm w t w t F w b B w b F/(-F/ ) B/(-B/ )

21 Volm Nomalizatio Tasfomatio V t V b V l V

22 Volm Nomalizatio Tasfomatio F/(-F/ ) B/(-B/ ) o t Fo ) ( ) ( ) ( ) )( ( 2 B F B F F B B F F F B B F F o o t + Scalig s Taslatio

23 s s s N ) ( ))/ )( (( ) )/( ( ) )/( ( 2 F B F B s w w s w w s b t b t l l Volm Nomalizatio Tasfomatio wh ) ( )/ ( ) )/( ( ) )/( ( B F B F w w w w w w w w b t b t t b l l l Total Tasfomatio: A w M s M N

24 Pili Assigmt 3D It Sc i WCS Sc with las/olygos Wold Coodiat Systm (WCS)

25 Pili Assigmt V U Ey VRP i WCS N Ey ad Widow xtds i VCS Viwig Coodiat Systm (VCS)

26 Pili Assigmt Ey Viw Pla, Fot Pla F Back Pla B Fot ad Back la i VCS Ey alog N-axis

27 Pili Assigmt w t w t F w b B w b F/(-F/ ) B/(-B/ )

28 Pili Assigmt V t V b V l V

29 Pili Assigmt Dislay of ocss of iwig tasfomatio sig OGL Th sc i wold Th sc (withot cliig) aft tasfomatio to VCS with iw fstm Th sc aft scti tasfomatio Th sc aft omalizatio Th sc aft ojctio ad cliig o th iwig la Chag of it aamts

30 Pili Assigmt Th sc i wold

31 Pili Assigmt Th sc aft tasfomatio to VCS with iw fstm

32 Pili Assigmt Th sc aft scti/omalizatio tasfomatio

33 Pili Assigmt Th sc aft ojctio ad cliig o th iwig la

34 Pili Assigmt