Tranformations. Some slides adapted from Octavia Camps

Transcription

1 Trformtio Some lide dpted from Octvi Cmp

2 A m 3 3 m m 3m m Mtrice 5K C c m ij A ij m b ij B A d B mut hve the me dimeio m

3 3 Mtrice p m m p B A C m k kj ik ij b c A d B mut hve A d B mut hve comptible dimeio comptible dimeio A B B A Idetit Mtri: A AI IA I

4 Mtrice Aocitive T*(U*(V*p)) = (T*U*V)*p Ditributive T*(u+v) = T*v + T*v 4

5 Mtrice Trpoe: C m c A T ij ji m ( A B) ( AB) T T B A T T A T B T If A T A A i mmetric 5

6 6 Mtrice,AJAH E = J,AJAH E = J ) KIJ >A IGK=HA ) KIJ >A IGK=HA det det

7 Mtrice Ivere: A mut be qure A A A A I 7

8 Euclide trformtio 8

9 D Trltio J 9

10 D Trltio Equtio JO J P (, ) O t ( t, t ) N JN P ' ( t, t ) P t

11 D Trltio uig Mtrice N O JN JO J ), ( ), ( t t t P ' t t t t P J

12 Sclig

13 3 Sclig Equtio N O I N I N I O I O ), ( ' ), ( P P P P ' P' S P S P '

14 Rottio 4

15 Rottio Equtio ; O + K JAH?? MEIA H J=JE >O = = C A : N ' co i ' i P' R.P co 5

16 Wh doe multiplig poit b R rotte them? Thik of the row of R ew coordite tem. Tkig ier product of ech poit with thee epree tht poit i tht coordite tem. Thi me row of R mut be orthoorml vector (orthogol uit vector). Thik of wht hppe to the poit (,) d (,). The go to (co thet, -i thet), d (i thet, co thet). The remi orthoorml, d rotte clockwie b thet. A other poit, (,b) c be thought of (,) + b(,). R((,)+b(,) = R(,) + R(,) = R(,) + br(,). So it i the me poitio reltive to the rotted coordite tht it w i before rottio reltive to the, coordite. Tht i, it rotted. 6

17 Degree of Freedom ' co i ' i co 4 EI N " A A A JI *76 6DAHA EI B BHAA@ 6DA " A A A JI KIJ I=JEIBO JDA B ME C? IJH=E JI R R T det( R) R T R I 7

18 Trformtio c be compoed Mtri multiplictio i ocitive. Combie erie of trformtio ito oe mtri. (emple, whitebord). I geerl, the order mtter. (emple, whitebord). D Rottio c be iterchged. Wh? 8

19 Rottio d Trltio ( )( co -i t i co t ) Rottio, Sclig d Trltio ( -b t )( b t ) 9

20 Rottio bout rbitrr poit C trlte to origi, rotte, trlte bck. (emple, whitebord). Thi i lo rottio with oe trltio. Ituitivel, mout of rottio i me either w. But trltio i dded.

21 Ivere of rottio If R i rottio, RR T = I. Thi i becue the digol of RR T re the mgitude of the row, which re ll, becue the row re uit vector givig directio. The off-digol re the ier product of orthogol uit vector, which re zero. So the trpoe of R i it ivere, rottio of equl mgitude i the oppoite directio.

22 Stretchig Equtio N O 5N N N 5O O O P' ), ( ' ), ( P P S P S P '

23 3 Stretchig = tiltig d projectig (with wek perpective) P'

24 4 Lier Trformtio d c b i co co i i co co i i co co i i co co i P' SVD

25 Affie Trformtio b t P' c d t 5

26 Viewig Poitio Epre world i ew coordite tem. If origi me, thi i doe b tkig ier product with ew coordite. Otherwie, we mut trlte. 6

27 Suppoe, for emple, we wt to hve the i how how we re fcig. We wt to be t (7,3), fcig i directio (ct,t). The i mut be orthogol, (-t,ct). If we wt to epre (,) i thi coordite frme, we eed to tke: (ct,t)*(- 7,-3), d (-t,ct)*(-7,-3). Thi i doe b multiplig b mtri with row (-t,ct) d (ct,t) 7

28 8 Simple 3D Rottio z z z... co i i co Rottio bout z i. Rotte, coordite. Leve z coordite fied.

29 R co i i co Full 3D Rottio co i i co co i i co A rottio c be epreed combitio of three rottio bout three e. T RR Row (d colum) of R re orthoorml vector. R h determit (ot -). 9

30 Ituitivel, it mke ee tht 3D rottio c be epreed 3 eprte rottio bout fied e. Rottio hve 3 degree of freedom; two decribe i of rottio, d oe the mout. Rottio preerve the legth of vector, d the gle betwee two vector. Therefore, (,,), (,,), (,,) mut be orthoorml fter rottio. After rottio, the re the three colum of R. So thee colum mut be orthoorml vector for R to be rottio. Similrl, if the re orthoorml vector (with determit ) R will hve the effect of rottig (,,), (,,), (,,). Sme reoig D tell u ll other poit rotte too. Note if R h determit -, the R i rottio plu reflectio. 3

31 3D Rottio + Trltio Jut like D ce 3

32 Trformtio of lie/orml D. Lie i et of poit (,) for which (,).(b) T =. Suppoe we rotte poit b R. We wt mtri, T, o tht: R*(,).T 3

33 3D Viewig Poitio Row of rottio mtri correpod to ew coordite i. 33

34 Rottio bout kow i Suppoe we wt to rotte bout u. Fid R o tht u will be the ew z i. u i third row of R. Secod row i thig orthogol to u. Third row i cro-product of firt two. Mke ure mtri h determit. 34