01 = Transformations II. We ve got Affine Transformations. Elementary Transformations. Compound Transformations. Reflection about y-axis

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1 Leure Se 5 Trnsformions II CS56Comuer Grhis Rih Riesenfel 7 Ferur We ve go Affine Trnsformions Liner Trnslion CS56 Comoun Trnsformions Buil u omoun rnsformions onening elemenr ones Use for omlie moion Use for omlie moeling CS56 3 Elemenr Trnsformions Sle Roe Trnsle Sher Refle S S R T T Sh Sh Rf Rf CS56 4 Refeion ou -is CS56 5 Refleion ou -is CS56 6 CS56

2 Refleion ou -is - CS56 7 Refleion ou -is CS56 8 Is Refleion Elemenr? Cn we effe refleion in n elemenr w? More elemenr mens sle sher roion rnslion. Refleion is Sle - 9 CS56 Emle:Move lok hns Emle:Move lok hns CS56 CS56 CS56

3 CS56 3 CS56 3 Emle:Move lok hns CS56 4 Emle:Move lok hns CS56 5 Clok Trnsformions Trnsle o Origin Move hn wih roion Move hn k o lok Do oher hn CS56 6 Clok Trnsformions 5 o where T R T T T R T T s CS56 7 Clok Trnsformions os os CS56 8 M [] [] [ ]

4 CS56 4 CS56 9 M [] [] Trnsle o Origin M o rnsle inervl [ ] [ ] [ ] CS56 M [] [] Normlie he inervl M o normlie inervl [ ] [ ] [ ] CS56 M [] [] T S CS56 Jus Look [ ] R This is homogeneous form for D T S CS56 3 M [] [-] [ ] - CS56 4 M [] [-] Trnsle ener of inervl o origin Normlie inervl o [-]

5 CS56 5 CS56 5 M [] [-] Susiue nlogous for : CS56 6 Now M [] [] Firs m [] o [] We lre i his Then m [] o [] CS56 7 Now M [] [] Sle [] - Then rnsle Th is in D homogeneous form: CS56 8 All Togeher: [] [] CS56 9 Now M Rengles u m v m u v m m CS56 3 Trnsformion in n v v u u v u m m m m where

6 CS56 6 This is he Viewor Trnsformion Goo for g ojes from one oorine ssem o noher This is wh we o wih winows n viewors 3 CS56 3 3D Trnsformions Sle Roe Trnsle Sher T T T Sh Sh Sh S S S R R R CS D Sle in S CS D Sle in S CS D Sle in S CS D Sle in S

7 CS56 7 CS56 37 Overll 3D Sle S CS56 38 Overll 3D Sle Sme in n : Wh is Posiive Roion in 3D? Si en of given is Look Origin CC Roion is in Posiive ireion 39 3D Posiive Roions CS56 4 3D Roion ou -is We hve lre one his: os os R CS56 4 3D Roion ou -is

8 3D Roion ou -is os R os 3D Roion ou -is CS D Roion ou -is Elemenr Trnsformions os R os Sle Roe Trnsle Sher Refle S S R R T T Sh Sh Rf Rf CS56 45 CS56 46 Consier n rirr 3D roion Wh is is inverse? Wn o roe ou rirr is : R Wh is is rnsose? Cn we onsruivel eluie his relionshi? CS CS56 8

9 Firs roe ou : R Then roe ou β : R β Now in he --lne Roe in he --lne β β CS56 49 CS56 5 Now erform roion ou is: R - Now ligne wih -is Now erform roion ou is: R - Now ligne wih -is 5 5 Then roe ou β: R β Then roe ou : R β Roe gin in he --lne Now o originl osiion of β CS56 53 CS56 54 CS56 9

10 We effee roion ou rirr is : R We effee roion ou rirr is : R R R β R β R R R Roion ou n rirr is Roion ou -is n e effee omosiion of 5 elemenr roions We show rirr roion s suession of 5 roions ou rinil es CS56 57 R R os os os β β β os β In mri erms os os R R os β β β os β R β R β os os R CS56 58 Similrl R R so Rell [ AB ] B A R os os os β β β os β Consequenl for A R M R A R M R euse os os R os β β β os β os os CS [ R M R ] [ M R] [ R ] R M R CS56 6 CS56

11 CS56 CS56 6 R S M SR M SR S R I follows irel h CS56 6 R R R os os os os β β β β R os os os os os os β β β β CS56 63 R R Consruivel we hve shown This will e useful ler CS D Trnslion in T CS D Trnslion in T CS D Trnslion in T

12 CS56 CS D Sher in -ireion Sh CS D Sher in -ireion Sh CS D Shers:lm rinil lne sher in oher DoFs Sh CS56 7 3D Sher in -ireion Sh CS56 7 3D Sher in -ireion Sh

13 CS56 3 Sh CS D Sher in -ireion e e e Sh CS D Sher in e e e Sh CS D Sher in f f f Sh CS56 77 Wh is Perseive? A mehnism for orring 3D in D True Perseive orresons o rojeion ono lne True Perseive orresons o n iel mer imge Mn Kins of Perseive Use Mehnil Engineering Crogrh Ar

14 CS56 4 CS56 79 Perseive in Ar Nïve wrong Egin Cuis unrelisi Esher Imossile elois lol roer Heroli non-lnr e CS56 8 True Perseive in D h CS56 8 True Perseive in D h h CS56 8 True Perseive in D This is righ nswer for sreen rojeion CS56 83 True Perseive in D En Trnsformions I Leure Se 5 84