Trnformion Ordered e of number:,,,4 Emple:,,z coordine of p in pce. Vecor If, n i i, K, n, i uni ecor
Vecor ddiion +w, +, +, + V+w w Sclr roduc,,
Inner do roduc α w. w +,.,. The inner produc i SCLR!. w,., w coα. w w Fir, we noe h if we cle ecor, we cle i inner produc. Th i, <,w> <,w>. Thi follow pre direcl from he definiion. Thi men h he emen <,w> w colph i rue if nd onl if i i he ce h when nd w re uni ecor, <,w> colph, becue: <,w> </,w/ w > w. So from now on, we cn ume h w, re uni ecor. Then, n emple, we cn conider he ce where w,. I follow from he definiion of coine h <,w> colph. We cn lo ee h king <,,> nd <,,> produce he, coordine of. Th i, if, nd, re n orhonorml bi, king inner produc wih hem gie he coordine of poin relie o h bi. Thi i wh he inner produc i o ueful. We ju he o how h hi i rue for n orhonorml bi, no ju, nd,. How do we proe hee properie of he inner produc? Le r wih he fc h orhogonl ecor he inner produc. Suppoe one ecor i,, nd WLOG,>. Then, if we roe h b 9 degree counerclockwie, we ll ge, -. Roing he ecor i ju like roing he coordine em in he oppoie direcion. nd,*,-. Ne, noe h if w + w w, hen *w *w+w *w + *w. For n w, we cn wrie i he um of w+w, where w i perpendiculr o, nd w i in he me direcion. So *w. *w w, ince *w/ w. Then, if we ju drw picure, we cn ee h co lph w *w *w.
Inner produc nd direcion w α Thi ell u h if i uni ecor nd w in h <,w> w coα. Thi i he projecion of w ono. I men h o ge o w, we go dince of <,w> in he direcion, nd hen ome dince in direcion orhogonl o. Mrice n m M n M n L L L O L m m m M nm Sum: C n m n m + Bn m c + b ij ij nd B mu he he me dimenion ij 4
Mrice roduc: C n p n mbm p m c ij k ik b kj nd B mu he compible dimenion n nbn n Bn nn n Ideni Mri: O O I I I O O O O O Mrice Trnpoe: T C m n n m c ij ji T T B + + T T B B T B T If T i mmeric 5
Eucliden rnformion D Trnlion 6
7 D Trnlion Equion + + +, ',, D Trnlion uing Mrice,, + + '
Scling Scling Equion., ', '. ' ' S S 8
Roion Roion Equion Couner-clockwie roion b n ngle Y X ' ' co in ' R. in co 9
Wh doe mulipling poin b R roe hem? Think of he row of R new coordine em. Tking inner produc of ech poin wih hee epree h poin in h coordine em. Thi men row of R mu be orhonorml ecor orhogonl uni ecor. Think of wh hppen o he poin, nd,. The go o co he, -in he, nd in he, co he. The remin orhonorml, nd roe clockwie b he. n oher poin,,b cn be hough of, + b,. R,+b, R, + R, R, + br,. So i in he me poiion relie o he roed coordine h i w in before roion relie o he, coordine. Th i, i roed. Degree of Freedom ' co ' in R i 4 elemen BUT! There i onl degree of freedom: The 4 elemen mu if he following conrin: R R T R de R in co T R I
Trnformion cn be compoed Mri muliplicion i ociie. Combine erie of rnformion ino one mri. In generl, he order mer. D Roion cn be inerchnged. Wh? Roion nd Trnlion co -in in co Roion, Scling nd Trnlion -b b
Inere of roion If R i roion, RR T I. Thi i becue he digonl of RR T re he mgniude of he row, which re ll, becue he row re uni ecor giing direcion. The off-digonl re he inner produc of orhogonl uni ecor, which re zero. So he rnpoe of R i i inere, roion of equl mgniude in he oppoie direcion. S. Sreching Equion S. ' ' S, ', S
Sreching iling nd projecing wih wek perpecie ' Liner Trnformion d c b in co co in in co co in in co co in in co co in ' SVD
4 ffine Trnformion ' d c b Thi i equilen o reching b n rbirr moun in n rbirr direcion, nd rnling. Soling for Trnformion wih hee correpondence. So we cn ole for he re reled b n ffine Trnformion : If., imge : nd in he oher, Le poin in one imge he coordine : u u u u u u u i i i i