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LinearLinear Transformation Transformation with CG & animationwith CG & animation
Ogose Ogose ShigekiShigekiKawaiKawai--JukuJuku
Tokyo, JapanTokyo, Japanhttp://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.
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2 1 Transformatioex n by
1 11.
−
original transformed
muffine.jar
(2,1)(-1,1)
Example of linear
OA 3OB OA' 3O2
i
B'2
ty
+ → +
(1,0)
(0,1)
Transformation of a photo bcos sin
ex2 )sin c
y s
. (o
Rθ θ
θθ θ
−=
original transformed
muffine.jar
Use a palette or type it in the field.
cos30 ,sin( )30° °( ,sin 30 c )os30° °−
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2 1 Animation of Rotation&Enlargement by
1 2ex3. F
− =
animation by rotation
current matrix is1 0
0 1
current matrix is2 1
1 2
−
is right-at-the-moment matrix. Which starts from the unit matrix and finishes as the target matrix.Curr matrixent
2. EigenVectors & Animation
Animation is useful for rotation - which has no real eigenvectors- , but it works even better for transformationswhich have real eigenvectors.
Next example has 2 eigenvectors.
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1 1 5 ,
0 02
10 10A A
=
=
5 0
For 0 2
A =
，
1 0eigenvectors : & eigenvalues :5&2 , respectively.
0 1
，
3 2For
1 4B =
，2 2
· 21 1
5 ,1 1 11
·BB− −
=
=
1 2eigenvectors : & eigenvalues :5&2 , respectively.
1 1−
，
Comparison of 2 transformations which have same eigenvalues.ex4.
5 0Transformation by . When the base is
1 0 an
0 02d
1A =
original transformed
animation by eigenvectors
(5,0)
(0,2)
muffine.jar
5
3 21 4
Transformation by . When the base is 1 0
and 0 1
B
=
When the base is (1,0) , transformation looks one and (0,1 of m) any.
(3,1)
(2,4)
1 0base is &
0 1
muffine.jar
1 2
3 21 4
Transformation by . When the 1 2
base is and 1 1
eB e−
= =
=
When the base is eigenvectors, transformation is easier to understand.
(1,1)(-2,1)
(5,5)
(-4,2)
1 2base is &
1 1−
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and have the same eigenvalues, thus they work similarly.Eigenvectors&values decide how linear transformations work.A B
by A
by B
by Bby Aby B
by Bby Aby 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
