Projective geometry- 2D. Homogeneous coordinates x1, x2,
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1 Projece geomer- D cknowledgemen Marc Pollefe: for allowng e ue of ecellen lde on opc p:// Rcard arle and ndrew Zerman "Mulple Vew Geomer n Compuer Von" omogeneou coordnae omogeneou repreenaon of lne a + b + c ( abc) ka ( abc ) ~ k( abc) ( ) + ( kb) + kc k equalence cla of ecor an ecor repreenae Se of all equalence clae n R () form P omogeneou repreenaon of pon ( ) on l ( abc) f and onl f a + b + c ( )( abc) ( ) l ( ) ~ k( ) k e pon le on e lne l f and onl f ll ( ) omogeneou coordnae Inomogeneou coordnae ( ) bu onl DOF Sprng 6 Projece Geomer D
2 Pon from lne and ce-era Inerecon of lne e nerecon of wo lne l and l' l l' Lne jonng wo pon e lne roug wo pon and ' l ' Eample Sprng 6 Projece Geomer D Ideal pon and e lne a nfn Inerecon of parallel lne l ( a b c) and l' ( a b c' ) l l' ( b a) Eample Ideal pon ( ) Lne a nfn l ( ) Noe a e le on a ngle lne P R l Noe a n P ere no dncon beween deal pon and oer Sprng 6 Projece Geomer D
3 Summar e e of deal pon le on e lne a nfn nerec e lne a nfn n e deal pon lne parallel o l alo nerec n e ame deal pon rrepece of e alue of c. In nomogeneou noaon a ecor angen o e lne. I orogonal o (a b) -- e lne normal. u repreen e lne drecon. e lne drecon are e deal pon are oer. --> lne a nfn can be oug of a e e of drecon of lne n e plane. Sprng 6 Projece Geomer D 5 model for e projece plane Pon repreened b ra roug orgn Lne repreened b plane roug orgn plane repreen lne a nfn eacl one lne roug wo pon eacl one pon a nerecon of wo lne Sprng 6 Projece Geomer D 6
4 Dual l l l l l' l ' Dual prncple: o an eorem of -dmenonal projece geomer ere correpond a dual eorem wc ma be dered b nercangng e role of pon and lne n e orgnal eorem Sprng 6 Projece Geomer D 7 Conc Cure decrbed b nd -degree equaon n e plane a + b + c + d + e + f a + b + c + d + e + f or omogenzed or n mar form C w a C b / d / b / c e / d / e / f 5DOF: { a : b : c : d : e : f } Sprng 6 Projece Geomer D 8
5 5 Sprng 6 Projece Geomer D 9 Conc p://ccn.camoun.bc.ca/~jbron/jbconc.m Sprng 6 Projece Geomer D Fe pon defne a conc For eac pon e conc pae roug f e d c b a or ( ) c f ( ) f e d c b a c c ackng conran eld
6 angen lne o conc e lne l angen o C a pon on C gen b lc l C Sprng 6 Projece Geomer D Dual conc lne angen o e conc C afe l * C l * In general (C full rank): C C C* : djon mar of C. Dual conc lne conc conc enelope Sprng 6 Projece Geomer D 6
7 Degenerae conc conc degenerae f mar C no of full rank e.g. wo lne (rank ) C lm + ml e.g. repeaed lne (rank ) m l C ll l Degenerae lne conc: pon (rank ) double pon (rank) Noe a for degenerae conc * * ( C ) C Sprng 6 Projece Geomer D Projece ranformaon Defnon: eorem: projec an nerble mappng from P o elf uc a ree pon le on e ame lne f and onl f ( )( )( ) do. mappng :P P a projec f and onl f ere e a non-ngular mar uc a for an pon n P repreened b a ecor rue a () Defnon: Projece ranformaon ' ' ' Sprng 6 Projece Geomer D or ' 8DOF projeccollneaonprojece ranformaonomograp 7
8 Mappng beween plane cenral projecon ma be epreed b (applcaon of eorem) Sprng 6 Projece Geomer D 5 Remong projece doron elec four pon n a plane w know coordnae ' + + ' ' ' ' + + ' '( + + ) + '( + + ) (lnear n j ) ( conran/pon 8DOF pon needed) Remark: no calbraon a all necear beer wa o compue (ee laer) Sprng 6 Projece Geomer D 6 8
9 ranformaon of lne and conc For a pon ranformaon ' ranformaon for lne l' - ranformaon for conc l C ' - C - ranformaon for dual conc * * C' C Sprng 6 Projece Geomer D 7 Doron under cener projecon Smlar: quare maged a quare. ffne: parallel lne reman parallel; crcle become ellpe. Projece: Parallel lne conerge. Sprng 6 Projece Geomer D 8 9
10 Sprng 6 Projece Geomer D 9 Cla I: Iomere (oame mercmeaure) co n n co ' ' ± orenaon preerng: orenaon reerng: ' R E I R R pecal cae: pure roaon pure ranlaon DOF ( roaon ranlaon) Inaran: leng angle area Sprng 6 Projece Geomer D Cla II: Smlare (omer + cale) co n n co ' ' ' R S I R R alo know a equ-form (ape preerng) merc rucure rucure up o mlar (n leraure) DOF ( cale roaon ranlaon) Inaran: rao of leng angle rao of area parallel lne
11 Cla III: ffne ranformaon ' a ' a a a ' R()( R ) DR() D 6DOF ( cale roaon ranlaon) non-oropc calng! (DOF: cale rao and orenaon) Inaran: parallel lne rao of parallel leng rao of area Sprng 6 Projece Geomer D Cla VI: Projece ranformaon ' P ( ) 8DOF ( cale roaon ranlaon lne a nfn) con non-omogeneou oer e plane Inaran: cro-rao of four pon on a lne (rao of rao) Sprng 6 Projece Geomer D
12 Sprng 6 Projece Geomer D con of affne and projece on lne a nfn + Lne a nfn become fne allow o obere anng pon orzon. Lne a nfn a a nfn bu pon moe along lne Sprng 6 Projece Geomer D Decompoon of projece ranformaon P S I K R RK + K de K upper-rangular decompoon unque (f coen >) co 5 n 5. n 5 co 5 Eample:
13 Oerew ranformaon Projece 8dof ffne 6dof Smlar dof Eucldean dof a a r r r r a a r r r r Concurrenc collnear order of conac (nerecon angenc nflecon ec.) cro rao Parallellm rao of area rao of leng on parallel lne (e.g mdpon) lnear combnaon of ecor (cenrod). e lne a nfn l Rao of leng angle. e crcular pon IJ leng area. Sprng 6 Projece Geomer D 5
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