SUMMARY. CS380: Introduction to Computer Graphics Quaternions Chapter 7. Min H. Kim KAIST School of Computing 18/04/06.

Transcription

1 CS380: Inroducion o Compuer Graphics Quaernions Chaper 7 Min H. Kim KAIST School of Compuing Hello World 3D SUMMARY 2 1

2 Modelview marix Modelview marix (MVM) E 1 O Describes he orienaion and posiion of he view and he orienaion and posiion of he objec O wih respec o he eye frame p = o c = w Oc = e e E 1 Oc The verex shader will ake hese verex daa and perform he muliplicaion E 1 Oc, producing he eye coordinaes used in rendering normalmarix() produces he inverse ranspose of he linear facor o ge uniform NMVM E 1 3 Maerial appearance (overview) Bidirecional reflecance disribuion funcion (BRDF) Simple reflecance model = diffuse + specular Here, all vecors are uni vecors cos(θ) = n l 4 2

3 Moion callback Move he objec Marix4 M = makexroaion(delay) * makeyroaion(delax); Marix4 A = makemixedframe(objrb, eyerb); objrb = domoowra(m, objrb, A); Move he eye objrb = domoowra(inv(m), eyerb, A); Perform ego moion objrb = domoowra(inv(m), eyerb, eyerb); NB For he eye moions, we inver he M so ha he mouse movemens produce he image movemens in more desired direcions. 5 Chaper 7 QUATERNION ROTATION 6 3

4 Menal model: Quaernion Why do we need quaernions? How can we muliply wo vecors! a and b,! resuling in c!?! b! c! a! b =! c! a! c =! b! a 1 such ha! a and! c are orhogonal and ha! b and! c are orhogonal. 7 Moivaion For animaion, we will wan o inerpolae beween frames in a naural way. We will sudy quaernions as alernaive o roaion marices: R = r 0 Laer we will add back in he ranslaions 8 4

5 RECAP: 3D Roaion Every roaion fixes an axis of roaion and roaes by some angle abou ha axis. Roaion around he z axis: b 1 b 1 b 2 b 2 b 3 b 3 x y z cosθ sinθ 0 sinθ cosθ 0 0 x y z 9 RECAP: 3D Roaion Roaion around he x axis cosθ sinθ 0 sinθ cosθ Roaion around he y axis cosθ 0 sinθ 0 sinθ 0 cosθ 10 5

6 RECAP: xyz-euler angle roaion Axis of roaion k = k x,k y,k z xyz-euler angle roaion marix k x 2 v + c k x k y v k z s k x k z v + k y s k y k x v + k z s k y 2 v + c k y k z v k x s k z k x v k y s k z k y v + k x s k z 2 v + c where c := cosθ, s := sinθ, v := 1 c. 11 Euler Roaions Problems Axis-angle Euler roaions: sequenially applying axis-angles along he x, y and z axes. Problems: Changing axes Gimbal lock problem! Roaion order (local ransformaion) 12 6

7 Invariance of a roaion Invariance is a propery of a class of mahemaical objecs ha remains unchanged when ransformaions of a cerain ype are applied o he objecs. An objec flying hrough space wih no forces acing on i has is cener of mass follow a sraigh line. Is orienaion spins along a fixed axis. This kind of orienaion saisfies boh lef and righ invariance. o 0 = w R 0 ω 0 ĉ0 13 Wha if having more han wo roaions? Inerpolaion of roaions: Desired objec frame roaion for ime=0 : o 0 = w R 0 Desired objec frame roaion for ime=1 : o 1 = w R 1 We wish o find a sequence of frames oα for α [0...1], ha naurally roaes from o o 0 o 1 = w R 0 ω 0 ĉ0 o 1 = w R 1 ĉ 1 o 0 ω

8 Bad ideas 1 Linear inerpolaion of marices! R α := (1 α )R 0 + (α )R 1 and hen se o α = w! R α Each basis vecor simply moves along a sraigh line In his case, he inermediae roaion marices R α are no 15 Bad idea 2 Facor boh R 0 and R 1 ino 3 wo-dimensional roaions, so-called XYZ Euler angles k 2 x v + c k x k y v k z s k x k z v + k y s k y k x v + k z s k 2 y v + c k y k z v k x s k z k x v k y s k z k y v + k x s k 2 z v + c These hree scalar values could each be linearly inerpolaed using α and used o generae inermediae roaions 16 8

9 Bad idea 2 No naural, No invarian o choice of he world frame. The roaion frame keeps changing! This is called lef invariance We need an inrinsic geomeric operaion (describable independen of coordinaes) 17 Wha we wan 1 Like o firs creae a single ransiion marix R 1 R 0 This marix can, as any roaion marix, be hough of as a roaion of some θ degrees abou some axis [k x,k y,k z ] Suppose we had a power operaor: (R 1 R 1 0 ) α which gave us a roaion abou [k x,k y,k z ] by αθ degrees insead. Then we could se R α := (R 1 R 1 0 ) α R 0 and se o α = w R α o α = w (R 1 R 1 0 ) α R

10 Resul This is a sequence of frames obained by more and more roaion abou a single axis Read righ o lef Correc sar and finish: w (R 1 R 1 0 ) 0 R 0 = w R 0 = o 0 w (R 1 R 1 0 ) 1 R 0 = w R 1 = o 1 The ransiion roaion fixes a unique axis This axis depends only on o 0 and o 1. No any choice of world frame. Up o cycles (which we can uniquify soon). This gives us a unique inerpolaion 19 Resul 20 10

11 Hard par 1 Hard par: facor R 1 R 0 ino is axis/angle form Main quaernion idea: is o keep rack of he axis and angle a all imes, bu in a way ha allows our manipulaions. This will allow us o do his inerpolaion I also could help in general wih avoiding numerical drif away from RBTs. 21 The represenaion A quaernion is 4 uple wih operaions Wrien: ω ĉ where ωĉ is a scalar, and is a coordinae vecor in 3D. A roaion of θ degree abou a uni lengh axis is presened as ˆk. cos Oddiy: he division by 2 will θ be needed o make he sin operaions work ou as needed θ 2 Because we are roaing no in 3D, bu in 4D, we jus roae θ only half way hrough! ω ĉ 22 11

12 Anipodes of quaernion (diamerically opposie o i) Noe ha a roaion of θ degrees abou he axis ˆk gives us he same quaernion. A roaion of θ + 4π degrees abou an axis ˆk also gives us he same quaernion A roaion of θ + 2π degrees abou an axis ˆk, which in fac is he same roaion, gives us he negaed quaernion So anipodes represen he same roaion ransformaion θ + 2π Bu heads up regarding cycles and power 23 Cycles 1 R 1 R 0 marix can, be hough of as a roaion of some θ + n2π degrees for any ineger n. No relevan for linear ransformaion on vecors, bu is relevan for inerpolaion The naural choice is o choose n such ha θ + n2π is minimal. This migh resul in a negaive roaional angle (he oher way)

13 Uni norm quas. == roaions Squared norm is sum of 4 squares. Any quaernion of he form cos θ 2 cos θ sin θ 2 2 x sin θ = sin θ 2 2 y sin θ 2 z has a uni form Conversely, any such uni norm quaernion can be inerpreed (along wih is negaion) as a unique roaion marix. ω ĉ 1= ω 2 + x 2 + y 2 + z 2, ĉ =[x, y,z] ˆk =1 25 Operaions Qua * qua muliply ω 1 ĉ 1 ω 2 ĉ 2 = (ω 1 ω 2 ĉ 1 ĉ 2 ) (ω 1 ĉ 2 + ω 2 ĉ 1 + ĉ 1 ĉ 2 ) Where and are he do and cross produc on 3 dimensional coordinae vecors. Correcly models (roaion marix) * (roaion marix) muliplicaions Example: M = Marix4::makeXRoaion(-dy) * Marix4::makeYRoaion(dx); 26 13

14 14 Uni norm quas Ideniy roaion example Flip roaion example Uni quaernion muliplicaion 27 1 ˆ0, 1 ˆ0 0 ˆk, 0 ˆk 0 ĉ 1 0 ĉ 2 = ĉ 1 ĉ 2 ĉ 1 ĉ 2 ˆk 1 ˆk 2 ˆk 1 ˆk 2 = 0 ˆk 2 0 ˆk 1 Operaions scalar * qua muliply Quaernions q and q are he same roaion!!! Uni quaernion muliplicaive inverse (conjugae) 28 α ω ĉ = αω αĉ, example) -1 ω ĉ = ω ĉ cos θ 2 sin θ 2 1 = cos θ 2 sin θ 2, 1 ˆ0 = cos θ 2 sin θ 2 cos θ 2 sin θ 2 1

15 RECAP: Affine roaion ransformaion Roae a vecor by an affine ransformaion for roaion: Sar wih 4-coordinae vecor c = ĉ 1 Lef muliply i by a 4 by 4 roaion marix o ge: c = Rc Wih resul of from R c = cˆ 1 29 Roae a vecor by a uni quaernion Le q be represened wih he uni norm quaernion: q = cos θ 2 sin θ 2 Conjugae of quaernion q Use a vecor Cvec3 o creae he non-uni norm quaernion ĉ 1 cos θ 2 q 1 = sin θ = 2 v = 0 ĉ cos θ 2 sin θ

16 Roae a vecor by a uni quaernion Perform he following riple quaernion muliplicaion: cos θ cos θ q 0 2 = 0 v' 2 cˆʹ sin θ ĉ sin θ 2 v 2 v'= qvq 1 We need inv(q) addiionally, such ha q and v are no orhogonal! If q and v are orhogonal, v =qv. ĉ 1 ĉ 2 (ω 1 ĉ 2 +ω 2 ĉ 1 +ĉ 1 ĉ 2 ) If we jus muliply q wih v, qv canno become a vecor (he firs elemen will be non-zero). The addiional righ muliplicaion of inverse allows us o ge a roaed vecor. Bu we will wrie his in code as: cvec = qua * cvec ω 1 ω 2 1 = (ω 1 ω 2 ĉ 1 ĉ 2 ) 31 Quaernions summary Expression: q = ω + xi + yj+ zk Muliplicaion rules: i 2 = j 2 = k 2 = ijk = 1 A conjugae (inverse) of a quaernion: q* = ω xi yj zk, q 1 = q * uni quaernion: q 2 = qq* = q *q =1, 1= ω 2 + x 2 + y 2 + z 2 A quaernion v, a vecer in 3D when w=0: v = v 1 i + v 2 j+ v 3 k Produc of he vecor wih he uni quaernion: v'q = qv v' = qvq 1 q

17 Quaernions inerpolaion To inerpolae beween wo frames relaed o world frame by R 0 and R 1 And suppose ha hese wo marices corresponds o he wo quaernions: cos θ 0 sin θ 0, 0 cos θ 1 sin θ 1 1 Linear inerpolaion (LERP): Simple, efficien approximaion Spherical linear inerpolaion (SLERP): More accurae way 33 LERP (Linear Inerpolaion) An even easier hack is o do 4D Linear inerpolaion (LERP) and renormalizaion a α p p = a +α(b a) p = a +α! v (1-α) p = (1 α)a +αb! v b cos θ 0 cos θ 1 (1 α ) sin θ 0 + α 0 sin θ

18 LERPing Boh lef and righ invarian. More efficien approximaion Useful for blending n differen roaions. cos θ 0 (1 α ) sin θ 0 0 cos θ 1 + α sin θ 1 1 NB his inerpolan is no longer a uni norm quaernion, i should be normalized afer calculaion. 35 SLERP (Spherical Linear Inerpolaion) This is called Spherical Linear inerpolaion (SLERP) or jus slerping since i happens o mach moving on a grea circle in 4 Power-based SLERP R α :=(R 1 R 0 1 ) α R

19 Power-based SLERP Spherical Linear Inerpolaion α p! p = a +α(b a)! v q v = rq 1 p = a +α v! r p = vq! = rq 1 q p = v! α q = rq 1 p = (1 α)a +αb cos θ! v = sin θ α v! = α θ = αθ 2 ( ) α q cos αθ! 2 v α = sin αθ 2 a α p (1-α)! v b 37 Power-based SLERP Firs, exrac he uni axis ˆk by normalizing he hree las enries of he quaernion. α Define cos θ sin θ 2 [ ] So we ge a unique value θ / 2 π...π hus a unique θ [ 2π...2π ] α = cos αθ 2 sin αθ 2 and As goes from 0 o 1, we ge a series of roaions wih angles 38 going beween 0 and θ 19

20 Power-based SLERP Bu wha if he ransiion quaernion cos θ 2 presens a of more han 180( ) sin θ θ π degrees : In paricular, if cos θ hen 2 < 0 θ π...2π So αθ would go more han 180 degrees which we don wan during inerpolaion In his case, suppose we had swapped o he anipode before calling power Then cos θ, we ge > 0 θ / 2 π / 2...π / 2 And hus θ π...π [ ] ω ĉ [ ] [ ] 39 Power-based SLERP cn(): 1 ω ĉ In order o selec he shor inerpolaion of less han 180 degrees, When we inerpolae, before calling he power operaor, we firs check he sign of he firs coordinae, and condiionally negae he quaernion. We call his he condiional negaion operaor Quaernions q and q are he same roaion!!! Finally, we oupu: cos θ 1 cos θ 0 cn sin θ 1 1 sin θ α cos θ 0 sin θ 0 0 = ω ĉ [ ω, ĉ] [ω,ĉ] 40 20

21 Sphere-based SLERPing In any dimension n, a rigonomeric argumen can be used o show ha spherical linear inerpolaion beween any wo uni vecors in! n, can be calculaed as: sin[(1 α )Ω] sin(ω) sin[(1 α)ω]! v sin(ω) 0 + sin[αω]! sin(ω) where Ω = cos 1 (! v 0! v 1 ) cos θ 0 sin θ sin[αω] sin(ω) v 1 cos θ 1 sin θ Puing back he ranslaion Les now build a daa srucure o represen an RBT Recall: RBT daa srucure Our class r Class RigTForm{ Cvec3 ; Qua r; }; = i A = TR r

22 RBT Inerpolaion Given wo frames o 0 = w O 0, o 1 = w O 1 Given wo RBTs We will wrie i as marices O 0 = (O 0 ) T (O 0 ) R and O 1 = (O 1 ) T (O 1 ) R, bu implemen in our RigTform daa ype. Inerpolae beween hem by: linearly inerpolaing he wo ranslaions o ge: Slerp beween he roaion quaernions o obain he roaion R α Se he inerpolaion RBT O α o be T α R α Se o α = w O α T α 43 RBT Inerpolaion Behavior Origin of o ravels in a sraigh line wih consan velociy, The vecor basis of o roaes wih consan angular velociy abou a fixed axis. Physically naural if origin is a cener of mass Time 44 22

23 RBT Inerpolaion Behavior Even hough he quaernion roaion is lef and righ invarian, he quaernion roaion + objec ranslaion is lef invarian. The ranslaion of he origin plays special role. If we use differen objec frames for same geomery, we ge differen inerpolaions No righ invarian 45 Code Change skyrb and objecrb[] o be RigTform daa ype insead of Marix4 In fac, almos all of he C++ Marix4s should ge replaced! We provide RigTForm makexroaion (cons double ang) You provide code for he produc of a RigTForm A and a Cvec4 c, o reurn A.r * c + Cvec4(A., 0). Wha if c has 0 fourh coordinae, hen no ranslaion should be done! Hin: v'= qvq

24 RBT * RBT Le us look a he produc of wo such rigid body ransforms. i 1 r 1 0 i 2 r 2 0 = i 1 r 1 r 1 2 r 2 0 = i 1 i r 1 2 r 1 0 r 2 0 = i 1 +r 1 2 r 1 r 2 0 A = TR 47 RBT * RBT The resul is a new rigid ransform wih ranslaion 1 + r 1 2 and roaion r 1 r 2 Use his o code up he * operaor. Mind he Cvec3s (he s) and Cvec4s (needed for q*v). 3D vecor i 1 + r 1 2 4D vecor r 1 r

25 Inverse RBT Likewise for inverse i r 0 r r 1 r 0 i i 1 r = = = = The resul is a new rigid body ransform wih ranslaion r 1 and roaion r 1 i r 1 A = TR r More code In GLSL, you will sill use is marix daa ype. The only Marix4s (ha will survive) are he projmarix, he MVM and he NMVM, which ge sen o your shaders. Also, when we need o do objec scaling, we canno capure his in an RigTform, so his will also be an Marix4 used in creaing he MVM. To communicae wih he verex shader using 4- by-4 marices, we need a procedure maketranslaion(rigtform) and Marix4 quatomarix (RigTform) o use T * R, which urns quaernions ino a 4-by-4 roaion marix

26 More code Then, he marix for a rigid body ransform can be compued as: Marix4 rigtformtomarix(cons RigTform& rb){ marix4 T = maketranslaion(rb.getranslaion()); marix4 R = quatomarix(rb.geroaion()); reurn T * R; } Thus our drawing code sars wih Marix4 MVM = rigtformtomarix(inv(eyerb) * objrb) \\ can righ muliply scales here Marix4 NMVM = normalmarix(mvm); sendmodelviewnormalmarix(curss, MVM,NMVM); 51 Uni Quaernion o Marix4 Quaernion (Qua) o Marix4 (R) cos θ sin θ x sin θ = ω y ĉ sin θ z = ω x y z 1 2y 2 2z 2 2xy 2ωz 2xz + 2ωy 0 2xy + 2ωz 1 2x 2 2z 2 2yz 2ωx 0 2xz 2ωy 2xz + 2ωx 1 2x 2 2y ω 2 + x 2 + y 2 + z 2 =

27 How o conver Qua o Marix4 Pseudo code inline Marix4 quatomarix(cons Qua& q) { Marix4 r; cons double n = norm2(q); cons double wo_over_n = 2/n; r(0, 0) -= (q(2)*q(2) + q(3)*q(3)) * wo_over_n; r(0, 1) += (q(1)*q(2) - q(0)*q(3)) * wo_over_n; r(0, 2) += (q(1)*q(3) + q(2)*q(0)) * wo_over_n; r(1, 0) += (q(1)*q(2) + q(0)*q(3)) * wo_over_n; r(1, 1) -= (q(1)*q(1) + q(3)*q(3)) * wo_over_n; r(1, 2) += (q(2)*q(3) - q(1)*q(0)) * wo_over_n; r(2, 0) += (q(1)*q(3) - q(2)*q(0)) * wo_over_n; r(2, 1) += (q(2)*q(3) + q(1)*q(0)) * wo_over_n; r(2, 2) -= (q(1)*q(1) + q(2)*q(2)) * wo_over_n; asser(isaffine(r)); reurn r; Min H. Kim (KAIST) } Foundaions of 3D Compuer Graphics, S. Gorler, MIT Press, More code we will no need any code ha akes a Marix4 and convers i o a Qua. scale will sill represened by a Marix4. (more laer) Useful reference: hps://