Implementing Transformations. Why Transforms? Primitive Transformations: Translation. Affine Transform: Definition M = Q x Q y Q z

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1 Why Tasfoms? Wat to aimate objects ad camea Taslatios Rotatios Sheas Ad moe.. Wat to be able to use pojectio tasfoms Implemetig Tasfomatios We use affie (liea) tasfoms Why? Ca be epeseted usig matices Ca be composed to fom complex tasfoms. Need to tasfom vetices oly, ad use (fast) scacovesio fo asteiatio. Ca do a lot (ot eveythig) with matices Use 3 3 ad 4 4 matices ITCS 3050:Game Egie Pogammig 1 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 2 Geometic Tasfomatios Pimitive Tasfomatios: Taslatio Affie Tasfom: Defiitio Q = M P + T Poit P (P x, P y, P ) is tasfomed ito Q(Q x, Q y, Q ) as follows: Q x Q y Q = ap x + dp y + gp + T x = bp x + ep y + hp + T y = cp x + fp y + kp + T Q x Q y = a d g b e h P x P y + T x T y Q c f k P T Q = M P + T ITCS 3050:Game Egie Pogammig 3 Geometic Tasfomatios Note: M = T = = I T x T y T Q x Q y = P x + T x P y + T y Q P + T (Idetity) Taslatio does ot fit withi a 3 3 matix epesetatio! ITCS 3050:Game Egie Pogammig 4 Geometic Tasfomatios

2 Pimitive Tasfomatios:Rotatio Pimitive Tasfomatios: Scale M = S x S y 0, T = [ ] 0 0 S Q x Q y = P xs x P y S y Q P S S x = S y = S : Uifom Scalig Else, Diffeetial Scalig ITCS 3050:Game Egie Pogammig 5 Geometic Tasfomatios M = cos si 0 Si Cos 0 T = [ ] Q = M P + T = P x Cos P y Si = P x Si + P y Cos = P Q x Q y Q Q y P y O R φ Execise: Deive the above 2D otatio matix. ITCS 3050:Game Egie Pogammig 6 Geometic Tasfomatios R Q Q x P P x Example: Tasfomig a Cicle Assume a cicle with oigi at (0,0,0) ad uit adius gltaslatef(8,0,0); RedeCicle(); gltaslatef(3,2,0); glscalef(2,2,2); RedeCicle(); gltaslatef(3,2,0) glscalef (2,2,2) Homogeeous Coodiate Repesetatio Motivatio Taslatio: Q = P + T (Tx, T y, T ) Rotatio: Q = R() P Scale: Q = S(Sx, S y, S ) P gltaslatef (8,0,0) Taslatio ivolves a vecto additio istead of a vecto-matix multiplicatio. Bette if all tasfoms use vecto-matix multiply. ITCS 3050:Game Egie Pogammig 7 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 8 Geometic Tasfomatios

3 Example: Homogeeous Coodiates i 2D Homogeeous Fom: Taslatio Add a thid coodiate w: P 2d = (x, y) = P H = (x, y, w) To covet fom P H to P 2D, poject to w = 1 (divide by w coodiate ad discad the 3d coodiate). w = 0 epesets poits at ifiity. W T H (T x, T y, T ) = T x T y T W = 1 Plae (x/w, y/w,1) P (x, y, w) Thus T x T y T P x P y P 1 P x + T x P = y + T y P + T 1 ITCS 3050:Game Egie Pogammig 9 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 10 Geometic Tasfomatios Homogeeous Fom: Rotatios Homogeeous Fom: Scale S x S S H (S x, S y, S ) = y S 0 Rotatio about,, Z R () = R x () = R y () = Cos Si 0 0 Si Cos Cos Si 0 0 Si Cos 0 Cos 0 Si Si 0 Cos 0 ITCS 3050:Game Egie Pogammig 11 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 12 Geometic Tasfomatios

4 Shea Tasfom Reflectio Tasfom 2D: 2D: M y = [ ] [ ] , M 0 1 x = 0 1 B B 3D: Shea (h > 0) (Q x Q y ) = (P x + hp y, gp x + P y ) = 1 h yx h x 0 h M sh = xy 1 h y 0 h x h y 1 0 [ ] [ ] 1 h Px g 1 P y whee h xy = shea alog axis due to, etc. ITCS 3050:Game Egie Pogammig 13 Geometic Tasfomatios 3D: (Reflectio about a plae) M = A C ITCS 3050:Game Egie Pogammig 14 Geometic Tasfomatios C A Affie Tasfom Iveses Tasfomatio About a Refeece Poit T 1 (T x, T y, T ) = T ( T x, T y, T ) x y M I geeal, S 1 (S x, S y, S ) = S(1/S x, 1/S y, 1/S ) Cos Si 0 0 R 1 () = R T Si Cos 0 0 = MM 1 = M 1 M = I Taslate (x, y, ) to the oigi. Tasfom. Ivet taslatio M = T (x, y, ).S(S x, S y, S ).T ( x, y, ) Fo otatio about a poit, eplace scale by a otatio tasfom. ITCS 3050:Game Egie Pogammig 15 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 16 Geometic Tasfomatios

5 Composig Affie Tasfoms If M 1, M 2,..., M ae affie tasfoms to be applied i successio to a poit P. Q = (M.(M 1...(M 2.(M 1.P )...) = (M.M 1...M 3.M 2.M 1 )P Fo tasfomig lage umbes of poits, the matix poduct (M 1.M 2.M 3...M 1.M ) eeds to be pefomed oly oce. M comp Tasfomig Diectio ectos Motivatio: Need to tasfom (vetex) omals, light diectio Matices used to tasfom poits, lies caot be used to tasfom diectio vectos. Must use the taspose of the ivese of geomety tasfomig matices. Taslatios do ot affect omals, ad otatios ae othogoal (R 1 = R T ) N = (M 1 ) T Oigial Icoect Coect poly omal M comp = T (T x, T y, T ).S(S x, S y, S ).R().T ( T x, T y ) 0 ITCS 3050:Game Egie Pogammig 17 Geometic Tasfomatios Scale by 0.5 alog ITCS 3050:Game Egie Pogammig 18 Geometic Tasfomatios The Eule Tasfom The Eule Tasfom (cotd) Gimbal lock, the loss of a degee of feedom may occu. Fo istace, h = 0, p = π/2adias, cos( + h) 0 si( + h) E(h, π/2, ) = si( + h) 0 cos( + h) 010 Eule Tasfom : a meas to descibe oietatios Give a view dow the egative Z axis, with up i the ad to the ight, E(h, p, ) = R ()R x (p)r y (h) with (h, p, ) efeig to the head, pitch, oll agles, ad E 1 = E T ITCS 3050:Game Egie Pogammig 19 Geometic Tasfomatios which is a fuctio of oly oe agle. Useful to extact the Eule agles fom a composite otatio matix, F = f 00 f 01 f 02 f 10 f 11 f 12 = R ()R x (p)r y (h) f 20 f 21 f 22 Expad RHS ad solve (details, Sectio 3.2.2) ITCS 3050:Game Egie Pogammig 20 Geometic Tasfomatios

6 Quateios: Defiitios Quateios Iveted by Si William Rowa Hamilto i 1843 Extesio of complex umbes Itoduced to compute gaphics by Ke Shoemake i 1985 A compact, efficiet meas to epesetig ad itepolatig oietatios. Supeio to both Eule agles ad matices, especially fo otatios. A quateio ˆq has 4 compoets, ˆq = (q v, q w ) = iq x + jq y + kq + q w = q v + q w i 2 = j 2 = k 2 = 1, jk = kj = i, ki = ik = j, ij = ji = k q w is the eal pat, ad q v the imagiay pat. All omal vecto opeatios (additio, scale, dot ad coss poduces) ca be pefomed o q v. Multiplicatio: ˆqˆ = (iq x + jq y + kq + q w )(i x + j y + k + w ) = (q v v + w q v + q w v, q w w q v v ) ITCS 3050:Game Egie Pogammig 21 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 22 Geometic Tasfomatios Quateios: Defiitios (cotd) Additio: ˆq + ˆ = (q v + v, q w + w ) Cojugate:ˆq = (q v, q w ) = ( q v, q w ) Nom: (ˆq) = ˆqˆq = ˆq ˆq = q v q v + q 2 w = q 2 x + q 2 y + q 2 + q 2 w Idetity: î = (0, 1) Multiplicative Ivese: (ˆq) 2 = ˆqˆq ˆqˆq (ˆq) = 1 2 ˆq 1 1 = (ˆq) 2 ˆq A quateio ˆq, with (ˆq) = 1. ˆq ca be witte as Give u q = 1, Uit Quateios ˆq = (siφu q, cosφ) = siφu q + cosφ ( ˆq) = si 2 φ(u q u q ) + cos 2 φ = 1 Quateios ae pefect fo ceatig otatios ad oietatios. Othe Popeties: Cojugate, Nom ules, Lieaity, Associativity, log, powe. ITCS 3050:Game Egie Pogammig 23 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 24 Geometic Tasfomatios

7 Rotatio About Axis Rotatio About Axis: Taditioal Method Goal: To otate the vecto by about, R = R(, ) R Uit quateios ca epeset ay 3D otatio ey compact, efficiet otatio To otate p about u q, put p = (p x p y p p w ) T ito a uit quateio, siφu q, cosφ Result: ˆq ˆpˆq 1 otates ˆp aoud u q by the agle 2φ adias Note: q 1 = ˆq. ITCS 3050:Game Egie Pogammig 25 Geometic Tasfomatios R Decompose ito ad, paallel ad pepedicula to = ( ) = ( ) ITCS 3050:Game Egie Pogammig 26 Geometic Tasfomatios Rotatio About Axis: Taditioal Method Rotatio About Axis: Taditioal Method R R R R Detemie R by computig, othogoal to ad : = Thus, R = (cos) + (si) ITCS 3050:Game Egie Pogammig 27 Geometic Tasfomatios Thus R = R + R = R + (cos) + (si) = ( ) + cos( ( )) + (si) = (cos) + (1 cos)( ) + (si) ITCS 3050:Game Egie Pogammig 28 Geometic Tasfomatios

8 Rotatio Usig Quateios T he same otatio, R = R(2, ) ca be pefomed usig the (uit) quateio poduct ˆq ˆpˆq 1 Assume ˆq = (q v, q w ) ˆq is a uit quateio ˆq = (si, cos), = 1 Give Compute pq 1 : Remembe Deivatio: ˆq ˆpˆq 1 p = (0, ) q = (s, v) q 1 = ˆq = (s, v) ˆp(s, v)ˆq(s, v ) = s 2 vv, v v + sv + s v Compute the poduct ˆq ˆpˆq 1. Ca be show to be ˆq ˆpˆq 1 = (q 2 w v v) + 2v(v ) + 2s(v, 0) = (cos 2 si 2 ) + 2si 2 ( ) + 2cossi( )) = (cos2 + (1 cos2)( ) + si2( ), 0) Thus, pq 1 = (0, )(s, v) = (0 + v), ( v) + s. ITCS 3050:Game Egie Pogammig 29 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 30 Geometic Tasfomatios Deivatio: ˆq ˆpˆq 1 (cotd) Compute q(pq 1 ): q(pq 1 ) = (s, v)( v, ( v) + s) = (s( v) v ( v) s(v )), (v ( v)) + s(v ) + s( v) + s 2 + ( v) v = (0, (v v) ( ) + (v ) v + s(v ) + s( v) + s 2 + ( v) v = (s 2 v v) + 2s(v ) + 2(v )v Fo uit quateios, s = cos, v = si, thus q(pq 1 ) = (cos 2 si 2 ) + 2si(si) + 2cos(si ) = cos(2) + (1 cos(2)) + si(2)( v) Idetical to the taditioal method, except fo a facto of 2, easily adjusted by edefiig q = (si 2, cos 2 ) ITCS 3050:Game Egie Pogammig 31 Geometic Tasfomatios Taditioal Rotatio vs Quateio Rotatio: Summay Expessio obtaied usig taditioal method: R = (cos) + (1 cos)( ) + (si) Expessio obtaied usig quateio otatio: R = (cos(2) + (1 cos(2))( ) + si(2)( )) ITCS 3050:Game Egie Pogammig 32 Geometic Tasfomatios

9 Quateios i Matix Fom 1 s(qy 2 + q) 2 s(q x q y q w q ) s(q x q + q w q y ) 0 M q = s(q x q y + q w q ) 1 s(qx 2 + q) 2 s(q y q q w q x ) 0 s(q x q q w q y ) s(q y q + q w q x ) 1 s(qx 2 + qy) 2 0 Fo uit quateios, s = 2/(ˆq), 1 2(qy 2 + q) 2 2(q x q y q w q ) 2(q x q + q w q y ) 0 M q = 2(q x q y + q w q ) 1 2(qx 2 + q) 2 2(q y q q w q x ) 0 2(q x q q w q y ) 2(q y q + q w q x ) 1 2(qx 2 + qy) 2 0 ITCS 3050:Game Egie Pogammig 33 Geometic Tasfomatios Rotatio fom Oe ecto to Aothe To otate fom vecto s to vecto t Method: Nomalie s ad t. Compute uit otatio axis, u = s t s t e = s t = cos(2), s t = si(2), 2φ, the agle betwee the vectos Quateio ˆq = (siφu, cosφ) With optimiatios (efe text) e + hv 2 x hv x v y v hv x v + v y 0 hv R(s, t) = x v y + v e + hvy 2 hv y v v x 0 hv x v v y hv y v + v x e + hv 2 0 whee h = 1 1 e ITCS 3050:Game Egie Pogammig 34 Geometic Tasfomatios Spheical Liea Itepolatio Smooth itepolatio of quateios, fom ˆq to ˆ. Useful i aimatig objects This is doe by the followig: ŝ(ˆq, ˆ, t) = (ˆˆq 1 ) tˆq, Algebaic Fom si(φ(1 t)) ŝ(ˆq, ˆ, t) = slep(ˆq, ˆ, t) = ˆq + si(φt) siφ siφ ˆ The secod fom is moe useful, effectively itepolatig ove a 4D sphee at fixed costat speed (geodesic itepolatio). Applicatio: Camea Positioig ad Oietatio Give camea positioed at (0, 0, 0) T ad lookig dow v = (0, 0, 1) T Goal: To move lookat diectio to w ad move camea to a ew positio p Accomplished by = T(p)R(v, w) Usually, will eed to otate the camea up diectio to somethig moe desiable. ITCS 3050:Game Egie Pogammig 35 Geometic Tasfomatios ITCS 3050:Game Egie Pogammig 36 Geometic Tasfomatios

10 Rotatio about a Abitay Axis:Methodology Rotatio about a Abitay Axis s y t x M Sometimes, useful to otate about a abitay axis, t s y Need two additioal axes to fom a basis, the chage bases. Chage fom stadad basis to ew basis, otate by give agle, ad tasfom back ITCS 3050:Game Egie Pogammig 37 Geometic Tasfomatios M T x s y t x Fid othoomal axes of basis: Detemiig s: Set smallest compoet of to eo, swap the emaiig two tems ad egate the fist, (0, x, y ) x <, x < s = (, 0, x ) y < x, y < ( y, x, 0) < x, < y s = s/ s t = s Rotatio to stadad basis: M = Fial Tasfom: = M T R x (α)m T s T t T ITCS 3050:Game Egie Pogammig 38 Geometic Tasfomatios