3D Geometrical Transformations. 3D Geometrical Transformations. 3D Coordinate Systems. Reminder: Cross Product. 3D Point Representation

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4 Chng o Coorins Ls hk h rnsormion o h X is nr - : Similrl, Y gos ino, n Z gos ino W oion Ao n Arirr Ais Ais o roion n lo n oin: 6.o.. ms si oins n h i: mk h is oinin ih on o h oorin s is, ro, n hn rnsorm k oion Ao n Arirr Ais s Iniil osiion rnsl o h Origin s o ono h Ais s 3 o h Oj Aron h Ais s 4 o h Ais o h originl Orinion s 5 rnsl o h Originl osiion oion Ao n Arirr Ais S: S : S 3: S 4: S 5: Comosiion: oion Ao Arirr Ais Consring n orhonorml ssm long h roion is: A or W rlll o h roion is: A or rnilr o W: A or orming righ-hn orhogonl ssm ih W n : s s ; s ; rnsorming lns ln rrsnion: B hr non-ollinr oins B imlii qion: A B C A B C C B A

5 rnsorming lns On o rnsorm ln is rnsorming n hr non-ollinr oins on h ln Anohr is o rnsorm h ln qion: Gin rnsormion sh h [,,, ] = [,,, ] in [A,B,C,], sh h: A B C rnsorming lns No h: A B C hs, h rnsormion h shol l o h ln qion is: A B C A B C

( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( )

( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( ) { } ( ) = ( ) = {,,, } ( ) β ( ), < 1 ( ) + ( ) = ( ) + ( ) max, ( ) [ ( )] + ( ) [ ( )], [ ( )] [ ( )] = =, ( ) = ( ) = 0 ( ) = ( ) ( ) ( ) =, ( ), ( ) =, ( ), ( ). ln ( ) = ln ( ). + 1 ( ) = ( ) Ω[ (

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