Transformations. Computer Graphics. Types of Transformations. 2D Scaling from the origin. 2D Translations. 9/22/2011. Geometric Transformation
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1 9// anfomaion. Compue Gaphic Lecue anfomaion Wha i a anfomaion? Wha oe i o? anfom he cooinae / nomal veco of objec Wh ue hem? Moelling -Moving he objec o he eie locaion in he envionmen -Muliple inance of a poope hape -Kinemaic of linkage/keleon chaace animaion Viewing Viual camea: paallel an pepecive pojecion Lecue Geomeic anfomaion anlaion oaion caling Linea peeve paallel line Non-unifom cale hea o kew ojecion peeve line epecive pojecion aallel pojecion Non-linea line become cuve wi ben wap moph pe of anfomaion Geomeic anfomaion Once he moel ae pepae we nee o place hem in he envionmen Objec ae efine in hei own local cooinae em We nee o anlae oae an cale hem o pu hem ino he wol cooinae em /9/ Lecue 4 oin efine a anlaeo oin a iance D anlaion. paallelo ai paallelo ai. Define he column veco Now. S Now D Scaling fom he oigin. oin efine a efom a cale ech o oin b a faco along he ai an along he ai.. Define he mai S o. 5 6
2 9// D oaion abou he oigin. D oaion abou he oigin..co.in 7 8 D oaion abou he oigin. D oaion abou he oigin..co.co.co.in.in.in.co.in.in.co.co.co.co.in.in.in.co.in.in.co.co.in Subiuing fo :.co.in Give u :.co.in.in.co 9 D oaion abou he oigin..co.in.in.co ewiing in mai fom give u : co in in. co co Define he mai in in co anfomaion. anlaion. Scale S oaion co in. in co We woul like all anfomaion o be muliplicaion o we can concaenae hem epe poin in homogenou cooinae.
3 9// Homogeneou cooinae A an ea cooinae W o a poin. W. wo e of homogeneou cooinae epeen he ame poin if he ae a muliple of each ohe. 5 an 46 epeen he ame poin. If W ivie b i o ge Caeian cooinae of poin /W/W. If W poin i ai o be a infini. anlaion in homogenie cooinae anfomaion maice fo D anlaion ae now. 4. Concaenaion. When we pefom anlaion on he ame poin 5 : So we epec Concaenaion. 6?. i : he mai pouc Mai pouc i vaioul efee o a compouning concaenaion o compoiion Concaenaion. 7 Mai pouc i vaioul efee o a compouning concaenaion o compoiion.. i : he mai pouc opeie of anlaion I Noe :. anlaion maice ae commuaive.
4 9// 4 Homogeneou fom of cale. 9 S ecall he fom of Scale : S In homogeneou cooinae : Concaenaion of cale.! mulipl iagonalelemenin he mai - ea o Onl. i: he mai pouc S S Homogeneou fom of oaion.. co in in co i.e : oaion maice ae ohogonal. Fo oaion maice Ohogonali of oaion maice. co in in co co in in co co in in co co in in co Ohe popeie of oaion. o be moe caeful nee Fo D oaion he ame i oaion of he ai onl becaue Bu hi i an I How ae anfom combine? //9 Lecue Scale anlae S Scale hen anlae Ue mai muliplicaion: p' S p S p Cauion: mai muliplicaion i NO commuaive!
5 9// Non-commuaive Compoiion Scale hen anlae: p' S p S p 5 Scale anlae anlae hen Scale: p' S p S p anlae 4 Scale 6 84 Non-commuaive Compoiion Scale hen anlae: p' S p S p S anlae hen Scale: p' S p S p S 6 //9 Lecue 4 5 //9 Lecue 4 6 How ae anfom combine? oae hen anlae q q oae 45 eg anlae anlae hen oae anlae /q/q oae 45 eg Cauion: mai muliplicaion i NO commuaive! //9 Lecue 4 7 Non-commuaive Compoiion oae hen anlae: p' p p - - anlae hen oae: p' p p - //9 Lecue pe of anfomaion. oaion an anlaion Angle an iance ae peeve Uni cube i alwa uni cube igi-bo anfomaion. oaion anlaion an cale. Angle & iance no peeve. Bu paallel line ae. anfomaion of cooinae em. Have been icuing anfomaion a anfoming poin. Alwa nee o hink he anfomaion in he wol cooinae em Someime hi migh no be o convenien i.e. oaing objec a i locaion 9 5
6 9// 6 anfomaion of cooinae em - Eample Concaenae local anfomaion maice fom lef o igh Can obain he local wol anfomaion mai p p p ae he wol cooinae of p afe each anfomaion anfomaion of cooinae em - eample i he wol cooinae of poin p afe n anfomaion Qui I a in he ca an fin he ie mio i.4m on m igh an.m in m fon I ae m ca an ove 5m fowa une egee o igh move 5m fowa again an une 45 egee o he igh an oppe Wha i he poiion of he ie mio now elaive o whee I wa iing in he beginning? Soluion he ie mio poiion i locall 4. he mai of fi iving fowa 5m i 4 5 Soluion he mai o un o he igh an 45 egee oaing - an -45 egee aoun he oigin ae 5 epecivel Soluion he local-o-global anfomaion mai a he la configuaion of he ca i he final poiion of he ie mio can be compue b p which i aoun
7 9// hi i convenien fo chaace animaion / oboic In oboic / animaion we ofen wan o know wha i he cuen D locaion of he en effeco like he han Can concaenae maice fom he oigin of he bo owa he en effece Define an j anfomaion of cooinae em. i Define M i M M a a poin in cooinae em i i j i a j k weobain b ubiuion : M i k I can alo behown ha : M j i i j k M M j i he anfom ha convea poin in em j o a poin in em i M j j j k 7 Lecue 4 8 D anfomaion. Ue homogeneou cooinae ju a in D cae. anfomaion ae now 44 maice. We will ue a igh-hane wol cooinae em - ou of page. Simple eenion o he D cae: anlaion in D. //9 Lecue Simple eenion o he D cae: Scale in D. oaion in D Nee o pecif which ai he oaion i abou. -ai oaion i he ame a he D cae. S co in in co 4 4 7
8 9// 8 oaing Abou he -ai 4 co in in co oaing Abou he -ai 44 co in in co oaion Abou he -ai 45 co in in co oaion abou an abia ai Abou u u u a uni veco on an abia ai 46 ' ' ' uu-cc uu-cu uu-c-u uu-c-u uu-cc uu-cu uu-cu uu-c-u uu-cc whee c co & in oaek u oaion No commuaive if he ai of oaion ae no paallel //9 α β β α Sheaing 48 a a
9 9// Calculaing he wol cooinae of all veice Fo each objec hee i a local-o-global anfomaion mai So we appl he anfomaion o all he veice of each objec We now know he wol cooinae of all he poin in he cene Nomal Veco We alo nee o know he iecion of he nomal veco in he wol cooinae em hi i going o be ue a he haing opeaion We onl wan o oae he nomal veco Do no wan o anlae i 49 5 Nomal Veco - We nee o e elemen of he anlaion pa o eo Viewing Now we have he wol cooinae of all he veice Now we wan o conve he cene o ha i appea in fon of he camea 5 5 View anfomaion We wan o know he poiion in he camea cooinae em We can compue he camea-o-wol anfomaion mai uing he oienaion an anlaion of he camea fom he oigin of he wol cooinae em Mw c View anfomaion We wan o know he poiion in he camea cooinae em vw Mw c vc oin in he wol cooinae Camea-o-wol anfomaion - vc Mw c vw oin in he camea cooinae Mc w vw Lecue
10 9// Summa. anfomaion: anlaion oaion an caling Uing homogeneou anfomaion D D anfomaion can be epeene b muliplicaion of a 44 mai Muliplicaion fom lef-o-igh can be coniee a he anfomaion of he cooinae em Nee o mulipl he camea mai fom he lef a he en eaing: Fole e al. Chape 5 Appeni ecion A o A5 fo eviion an fuhe backgoun Chape 5 55
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