Linear Transformation Transformation Linear with CG & - PDF document

Linear Transformation Transformation Linear with CG & animation with CG & animation Ogose Shigeki Ogose Shigeki Kawai- -Juku Juku Kawai Tokyo, Japan Tokyo, Japan http://mixedmoss.com/atcm/2012/ http://mixedmoss.com/atcm/2012/ 1. Advantages of using Computer Graphics (CG). • Grids can be drawn easily. • Effects of changing the ‘ original objects ’ or ‘ matrix ’ can be seen immediately. • Exotic objects such as photos can be transformed. • Animations can be used.

Example of linear i ty −   2 1 ��� � ��� � ���� � ���� � ex 1. Transformatio n by 1   + → + 2 OA 3OB 2 OA' 3O B' 1   muffine.jar (0,1) (-1,1) (2,1) (1,0) original transformed θ − θ   cos sin θ = ex2 . Transformation of a photo b y R ( )   θ θ sin c o s   muffine.jar Use a palette or type it in the field. ° ° ( cos30 ,sin 30 ) − ° ° ( sin30 c , os30 ) original transformed

−  2 1  =  ex3. Animation of Rotation&Enlargement by F  1 2   animation by rotation current matrix is  1 0  0   1   current matrix is −  2 1  1   2   Curr ent matrix is right-at-the-moment matrix. Which starts from the unit matrix and finishes as the target matrix. 2. EigenVectors & Animation Animation is useful for rotation - which has no real eigenvectors- , but it works even better for transformations which have real eigenvectors. Next example has 2 eigenvectors.

ex4. Comparison of 2 transformations which have same eigenvalues.  5 0    1   1 A   0   0 =  = = For A A 5 , 2 ，          0 2 0 0 1 1             1   0 eigenvectors : & ， eigenvalues :5&2 , respectively.     0 1     − −  3 2          1 1 2 2 =  = = For B ， B 5 · , B 2 ·          1 4 1 1 1 1           −   1  2  eigenvectors : & ， eigenvalues :5&2 , respectively.     1 1      5 0    1   0 =  Transformation by A . When the base is an d 1      0 2 0       muffine.jar animation by eigenvectors (0,2) (5,0) original transformed

. When the base is 1     0  3 2  = Transformation by B and       0 1  1 4      muffine.jar (2,4) (3,1)   1   0 base is &     0 1     When the base is (1,0) and (0,1 ) , transformation looks one of m any. � � �� −       1 2 3 2 =  = = Transformation by B . When the base is e and e      1 2 1 1  1 4      (5,5) (-4,2) (-2,1) (1,1) −   1  2  base is &     1 1     When the base is eigenvectors, transformation is easier to understand.

A and have the same eigenvalues, thus they work similarly. B Eigenvectors&values decide how linear transformations work. by A by B by A by B by A by B by A by B − − P BP 1 shows the similarity between A&B. P BP 1 represents � � �� the same transformation as , but it's the expression by B e and e . 1 2 Choose (1/P)AP Choose the base here. e1 & e2 are the base set by you. (left)