Transformations and Matrices Transformations I Transformations are - - PDF document

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Transformations I

CS5600Computer Graphics

by

Rich Riesenfeld 27 February 2002

Lecture Set 5

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Transformations and Matrices

• Transformations are functions
• Matrices are functions representations
• Matrices represent linear transf’s
• 2 x 2 Matrices == 2D Linear Transf’s

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What is a 2D Linear Transf ?

Recall from Linear Algebra: Definition: T(ax + y) = aT(x) + T(y) for scalar a and vectors x and y

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Example: Scale in x

Scale in x, by 2: (2 (xo + x1), y0 + y1) = (2xo, yo) + (2x1, y1)

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Example: Scale in x by 2

What is the graphical view?

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Scale in x by 2

y (x1, y1) (2 x1, y1) (x0, y0) (2 x0, y0) x

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( 2xo + 2x1, y0 + y1 )

y x

( 2 xo + 2 x1, y0 + y1 )

(2 x1, y1) (2 x0, y0)

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( 2 (xo + x1), y0 + y1 )

(x1, y1) (x0, y0) y x

(( xo + x1 ), y0 + y1 ) ( 2 (xo + x1), y0 + y1 )

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( 2 (xo + x1), y0 + y1 )

(( xo + x1 ), y0 + y1 ) ( 2 (xo + x1), y0 + y1 )

y x

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Summary on Scale

• “Scale then add,” is same as
• “Add then scale”

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Matrix Representation

Scale in x by 2:

2 0 0 1 x y

= 2x

y

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Matrix Representation

Scale in y by 2:

1 0 0 2 x y

= x

2y

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Matrix Representation

Overall Scale by 2:

2 0 0 2 x y

= 2x

2y

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Matrix Representation Showing Same

2 0 0 1 x0 + x1 y0 + y1 = 2(x0 + x1) y0 + y1

=

2x0 + 2x1 y0 + y1

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What about Rotation?

Is it linear?

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Rotate by θ

y x

(0,1)

θ

θ

(1,0)

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Rotate by θ: 1st Quadrant

y x

(cos θ , sin θ ) sin θ cos θ θ

(1,0)

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Rotate by θ: 1st Quadrant

(1, 0) (cos θ , sin θ )

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Rotate by θ: 2nd Quadrant

(0,1)

θ

θ

(1,0)

y x

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Rotate by θ: 2nd Quadrant

y

(0,1)

x

sin θ cos θ θ θ

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Rotate by θ: 2nd Quadrant

(0, 1) (-sin θ , cos θ )

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Summary of Rotation by θ

(0, 1) (-sin θ , cos θ ) (1, 0) (cos θ , sin θ )

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Summary (Column Form)

       − ⇒         θ θ cos sin 1

        ⇒         θ θ sin cos 1

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Using Matrix Notation

      =             θ θ θ θ θ θ sin cos 1 cos sin sin

• cos

      =             θ θ θ θ θ θ cos sin

• 1

cos sin sin

• cos

(Note that unit vectors simply copy columns)

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General Rotation by Matrix

      + =             θ θ θ θ θ θ θ cos sin ysin

• cos

cos sin sin

• cos

y x x y x

θ

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Who had linear algebra?

Who understand matrices?

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What do the off diagonal elements do?

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Off Diagonal Elements

      + =             y bx x y x b 1 1       + =             y ay x y x a 1 1

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Example 1

y       + =             = y x x y x y x T 4 . 1 4 . 1 ) , (

) 1 , ( ) , 1 (

) 1 , 1 (

) , (

S

x

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Example 1

x

y       + = y x x y x T 4 . ) , (

S

) 1 , ( ) , 1 ( ) 1 , 1 ( ) , (

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Example 1

y

) 1 , (

) 4 . , 1 (

) , (

T(S)

      + = y x x y x T 4 . ) , (

) 4 . 1 , 1 (

x

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Example 2

y       + =             = y y x y x y x T 6 . 1 6 . 1 ) , (

S

x

) 1 , ( ) , 1 ( ) 1 , 1 ( ) , (

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Example 2

x

y       + = y y x y x T 6 . ) , (

S

) 1 , ( ) , 1 ( ) 1 , 1 ( ) , (

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Example 2

x

y

) 1 , ( ) , 1 ( ) , (

T(S)

) 1 , 6 . (

      + = y y x y x T 6 . ) , (

) 1 , 6 . 1 (

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Summary

Shear in x: Shear in y:

      + =             = y ay x y x a Shx 1 1       + =             = y bx x y x b Shy 1 1

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Double Shear

      + =             ab) ( 1 1 1 1

1 1

b a a b       + =            

1 1

ab) ( 1 1 1 1 b a b a

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Sample Points: unit inverses

      =       −       1 1 1 1 b b       =      −       1 1 1

1

a a

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Geometric View of Shear in x

) , 1 (

) 1 , ( a −

) 1 , ( ) , 1 ( ) 1 , 1 (

Another Geometric View of Shear in x

39

x x

y y

Another Geometric View of Shear in x

40

x

y

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Geometric View of Shear in y

) 1 , ( ) , (

) , 1 ( b −

) 1 , ( ) , 1 ( ) , (

) 1 , 1 (

Another Geometric View of Shear in y

h h

42x

y

x

y

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Another Geometric View of Shear in y

43

x

y

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“Lazy 1”

             

1 1 1

                           

= 1 1 y x y x

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Translation in x

             

1 y x

                           

+ = 1

1 1 1

y x d x x d

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Translation in x

             

1 y x

                           

+ = 1

1 1 1

y d y x d y

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Homogeneous Coordinates

             

1 1 1

                           

= 1 1 y x y x

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Homogeneous Coordinates for , ≠ ↔

                     

=

λ

λ λ λ

y x

y x

             

1 y x

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Homogeneous Coordinates

Homogeneous term effects overall scaling

0, For ≠ λ

                                                                                 

↔

= = y x y x y x y x λ λ λ λ

λ λ

1 1 1 1

1 1

Homogeneous Coordinates

An infinite number of points correspond to (x,y,1). They constitute the whole line (tx,ty,t).

x w

y

w = 1 (tx,ty,t) (x,y,1)

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We’ve got Affine Transformations

Linear + Translation

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Next Class: Compound Transformations

• Build up compound transformations

by concatenating elementary ones

• Use for complicated motion
• Use for complicated modeling