Linear Transformation Transformation Linear with CG & - - PDF document

Linear Linear Transformation Transformation with CG & animation with CG & animation

Ogose Ogose Shigeki Shigeki

Kawai Kawai-

• Juku

Juku 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.

2 1 Transformatio ex n by 1 1 1. −      

• riginal

transformed

muffine.jar

(2,1) (-1,1)

Example of linear OA 3OB OA' 3O 2 i B' 2 ty + → +

• (1,0)

(0,1)

Transformation of a photo b cos sin ex2 ) sin c y s . (

• R

θ θ θ θ θ −  =     

• riginal

transformed

muffine.jar

Use a palette or type it in the field. cos30 ,sin ( ) 30

° °

( , sin30 c )

• s30

° °

−

2 1 Animation of Rotation&Enlargement by 1 2 ex3. F −   =    

animation by rotation

current matrix is 1 1       current matrix is 2 1 1 2 −       is right-at-the-moment matrix. Which starts from the unit matrix and finishes as the target matrix. Curr matrix ent

• 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.

1 1 5 , 2 1 1 A A    =           =           5 For 2 A   =     ，

1 eigenvectors : & eigenvalues :5&2 , respectively. 1             ，

3 2 For 1 4 B   =     ，

2 2 · 2 1 1 5 , 1 1 1 1 · B B − −     =     =                

1 2 eigenvectors : & eigenvalues :5&2 , respectively. 1 1 −             ，

Comparison of 2 transformations which have same eigenvalues.

ex4. 5 Transformation by . When the base is 1 an 2 d 1 A     =              

• riginal

transformed

animation by eigenvectors

(5,0) (0,2)

muffine.jar

3 2 1 4

Transformation by . When the base is 1 and 1 B                   =

When the base is (1,0) , transformation looks one and (0,1

• f m

) any.

(3,1) (2,4)

1 base is & 1            

muffine.jar 1 2

3 2 1 4

Transformation by . When the 1 2 base is and 1 1 e B e −     = =         =      

• When the base is eigenvectors, transformation is easier to understand.

(1,1) (-2,1) (5,5) (-4,2)

1 2 base is & 1 1 −            

and have the same eigenvalues, thus they work similarly. Eigenvectors&values decide how linear transformations work. A B

by A by B by B by A by B by B by A by A Choose the base here. e1&e2 are the base set by you. (left)

1 2 1 1

shows the similarity between A&B. represents the same transformation as , but it's the expression by and . P BP P BP B e e

− −

• Choose

(1/P)AP