CS-184: Computer Graphics. Today. Lecture #5: 3D Transformations and Rotations. 05-3DTransformations.key - September 21, 2016

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1 1 CS-184: Compute Gaphics Lectue #5: D Tansfomations and Rotations Pof. James O Bien Univesity of Califonia, Bekeley V016-S Today Tansfomations in D Rotations Matices Eule angles Eponential maps Quatenions

2 D Tansfomations Geneally, the etension fom D to D is staightfowad Vectos get longe by one Matices get eta column and ow SVD still woks the same way Scale, Tanslation, and Shea all basically the same Rotations get inteesting Tanslations à = 4 10t 01t y t à = 6010t y t z Fo D Fo D 4 4

3 5 Scales à = 4 s s y s à = 60 s y s z Fo D Fo D (Ais-aligned!) 5 Sheas à = 4 1 h y 0 h y h y h z 0 à = 6h y 1 h yz 0 7 4h z h zy Fo D Fo D (Ais-aligned!) 6 6

4 7 Sheas 1 h y h z 0 Ã = 6h y 1 h yz 0 7 4h z h zy Sheas y into Rotations 8 D Rotations fundamentally moe comple than in D D: amount of otation D: amount and ais of otation -vs- D D 8

5 9 Rotations 9 Rotations still othonomal Det(R)=16= 1 Peseve lengths and distance to oigin D otations DO NOT COMMUTE! Right-hand ule Unique matices DO NOT COMMUTE! Ais-aligned D Rotations 10 D otations implicitly otate about a thid out of plane ais 10

6 11 Ais-aligned D Rotations 11 D otations implicitly otate about a thid out of plane ais apple cos(θ) R = sin(θ) sin(θ) cos(θ) cos(θ) sin(θ) 0 R = 4sin(θ) cos(θ) Note: looks same as R Ais-aligned D Rotations 1 R = cos(θ) sin(θ) 5 ˆ 0 sin(θ) cos(θ) cos(θ) 0 sin(θ) R = ŷ sin(θ) 0 cos(θ) cos(θ) sin(θ) 0 R = 4sin(θ) cos(θ) 05 ẑ Z is in you face ẑ ŷ ˆ 1

7 1 Ais-aligned D Rotations 1 R = cos(θ) sin(θ) 5 ˆ 0 sin(θ) cos(θ) cos(θ) 0 sin(θ) R = ŷ sin(θ) 0 cos(θ) cos(θ) sin(θ) 0 R = 4sin(θ) cos(θ) 05 ẑ Also ight handed Zup ẑ ŷ ˆ Ais-aligned D Rotations 14 Also known as diection-cosine matices R = cos(θ) sin(θ) 5 R = 4 ˆ ŷ 0 sin(θ) cos(θ) cos(θ) sin(θ) 0 R = 4sin(θ) cos(θ) 05 ẑ cos(θ) 0 sin(θ) sin(θ) 0 cos(θ) 5 14

8 15 Abitay Rotations 15 Can be built fom ais-aligned matices: R = Rẑ Rŷ R ˆ Result due to Eule... hence called Eule Angles Easy to stoe in vecto But NOT a vecto. R = ot(,y,z) Abitay Rotations 16 R = Rẑ Rŷ R ˆ R ˆ Rŷ Rẑ R 16

9 17 Abitay Rotations 17 Allows tumbling Eule angles ae non-unique Gimbal-lock Moving -vs- fied aes Revese of each othe Eponential Maps 18 Diect epesentation of abitay otation AKA: ais-angle, angula displacement vecto Rotate θdegees about some ais Encode θ by length of vecto θ = ˆ θ 18

10 19 Eponential Maps Given vecto, how to get mati R Method fom tet: 1. otate about ais to put into the -y plane. otate about z ais align with the ais. otate θ degees about ais 4. undo # and then #1 5. composite togethe 19 Eponential Maps 0 Vecto epessing a point has two pats does not change otates like a D point 0

11 1 Eponential Maps 1 ` = ˆ 0 θ? = ˆ (ˆ )? 0 = + ` sin(θ)+? cos(θ) ` sin(θ)? cos(θ) Eponential Maps Rodiguez Fomula!! 0 = ˆ(ˆ ) +sin(θ)(ˆ ) cos(θ)(ˆ (ˆ )) Linea in Actually a mino vaiation...

12 Eponential Maps Building the mati 0 =((ˆˆ t )+sin(θ)(ˆ ) cos(θ)(ˆ )(ˆ )) (ˆ )= 4 0 ˆ z ˆ y ˆ z 0 ˆ 5 ˆ y ˆ 0 Antisymmetic mati (a )b = a b Easy to veify by epansion Eponential Maps 4 Allows tumbling No gimbal-lock! Oientations ae space within π-adius ball Nealy unique epesentation Singulaities on shells at π Nice fo intepolation 4

13 Eponential Maps 5 Why eponential? ( /n)ˆ 0 Instead of otating once by θ, let s do n small otations of θ/n Now the angle is small, so the otated is appoimately +( /n)ˆ = I + (ˆ ) n Do it n times and you get 0 = I + (ˆ ) n n 5 Eponential Maps 0 = lim I + (ˆ ) n n!1 n Remind you of anything? lim 1+ a n is a definition of n!1 n e a 6 So the otation we want is the eponential of (ˆ )! In fact you can just plug it into the infinite seies... 6

14 7 Eponential Maps 7 Why eponential? Recall seies epansion of e e = 1 + 1! +! +! + Eponential Maps 8 Why eponential? Recall seies epansion of e Eule: what happens if you put in iθ fo e iθ = 1 + iθ 1! + θ! + iθ + θ4! 4! + = 1 + θ! + θ4 θ 4! + + i 1! + θ! + = cos(θ)+isin(θ) 8

15 9 Eponential Maps 9 Why eponential? e (ˆ )θ = I + (ˆ )θ 1! + (ˆ ) θ! + (ˆ ) θ! + (ˆ )4 θ 4 + 4! But notice that: (ˆ ) = (ˆ ) e (ˆ )θ = I + (ˆ )θ 1! + (ˆ ) θ! + (ˆ )θ! + (ˆ ) θ 4 + 4! Eponential Maps 0 e (ˆ )θ = I + (ˆ )θ 1! θ e (ˆ )θ =(ˆ ) 1! + (ˆ ) θ! + (ˆ )θ! θ! + + I +(ˆ ) + θ! + (ˆ ) θ 4 + 4! θ 4 4! + e (ˆ )θ =(ˆ )sin(θ)+i +(ˆ ) (1 cos(θ)) 0

16 1 Quatenions 1 Moe popula than eponential maps Natual etension of e iθ = cos(θ)+isin(θ) Due to Hamilton (184) Inteesting histoy Involves hemaphoditic monstes Quatenions Ube-Comple Numbes q =(z 1,z,z,s)=(z,s) q = iz 1 + jz + kz + s i = j = k = 1 ij= k ji= k jk = i kj= i ki = j ik = j

17 Quatenions Multiplication natual consequence of defn. q p =(z q s p + z p s q + z p z q, s p s q z p z q ) Conjugate q =( z,s) Magnitude q = z z + s = q q Quatenions 4 Vectos as quatenions v =(v,0) Rotations as quatenions =(ˆsin θ Rotating a vecto,cos θ ) 0 = Composing otations = 1 Compae to Ep. Map 4

18 5 Quatenions 5 No tumbling No gimbal-lock Oientations ae double unique Suface of a -sphee in 4D = 1 Nice fo intepolation Intepolation 6 6

19 7 Rotation Matices 7 Eigen system One eal eigenvalue Real ais is ais of otation Imaginay values ae D otation as comple numbe Logaithmic fomula (ˆ )=ln(r)= θ sinθ (R RT ) T(R) 1 θ = cos 1 Simila fomulae as fo eponential... Rotation Matices 8 Conside: & $ RI = $ $ % y z Columns ae coodinate aes afte (tue fo geneal matices) Rows ae oiginal aes in oiginal system (not tue fo geneal matices) y yy zy z #& 1! $ yz! $ 0! " $ zz % # 0!! 1! " 8

20 9 9 Scene Gaphs 40 Daw scene with pe-and-post-ode tavesal Apply node, daw childen, undo node if applicable Nodes can do petty much anything Geomety, tansfomations, goups, colo, switch, scipts, etc. Node types ae application/implementation specific Requies a stack to implement undo post childen Nodes can cache thei childen Instances make it a DAG, not stictly a tee Will use these tees late fo bounding bo tees 40

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