2D modeling Transformation, Lecture notes of Computer Graphics and Animation

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Computer Graphics

Lec

2D Transformations

1

Transformations

What is a

? Transformation

2 Free powerpoint template: www.brainybetty.com

A transformation is a change in the position

or size of an object.

Movements that do not change the size or

shape of the object moved are called “rigid

transformations”.

 There are three types of transformations:

Translations , Scaling , and Rotations.

To learn more about each, check this out!

4

2D-Transformations

 Basic Transformations

 Homogeneous coordinate

system

 Composition of

transformations 5

2D-Transformations

Translation – 2D

7

x’ = x + dx y’ = y + dy

(4,5) (7,5)

Y

Before Translation X







































  

















 



 ^  

  

   

  

 

  ^  

  

 

1

0 0 1

0 1

1 0

1

Homogenious Form

y

x

d

d

y

x

P P T d

d T y

x P y

x P

y

x

y

x

(7,1) (10,1) X

Y

Translation by (3,-4)

 

  

 

 ^  

  

  

  



y

x y d

x d

y

x

0 1

1 0

Rotation – 2D

rotated

original

 

 

 

 

sin

cos

r

r v

  

   

      cos sin sin cos

cos cos sin sin expand y r r

x r r

sin cos

cos sin

sin

cos but y x y

x x y

y r

x r

 

sin

cos

r

r v

Scaling – 2D

10

: Types of Scaling

Differential ( sx != sy ) 

(4,5) (7,5) Uniform ( sx = sy )^ 

Y

X

(2,5/4) (7/2,5/4) X

Y

Before Scaling Scaling by (1/2, 1/4)

^ 

^ 

y

x

y

x

y

x

y s

x s

y

x s

s

S P P

y s y

x s x

Homogenious Form

y

x

y

x

s

s

x

x

Application of scaling “Reflection”

11 





(^) 









(^) 



 

    

0 0 1

0 1 0

1 0 0

Reflection about X- axis

M x

x x y y

^ 

Reflection about Y- axis

M y

x x y y

(1,1)

(1,-1)

Y

X

(-1,1) (1,1)

X

Y

Inverse 2D -

Transformations

13

y

- y

x

- x

( , )

- (sx,sy)

(-θ

- (θ

(-dx,-dy)

- (dx,dy)

M M

M M

S S

R R

T T

sx sy











1

1

1

)

1 )

1

Mirror Ref :

Sclaing :

Rotation :

Translaito n :

1 1

Homogeneous Co-

ordinates

14

Translation, scaling and rotation are

expressed (non-homogeneously) as: translation: P = P + T Scale: P = S · P Rotate: P = R · P

Composition is difficult to express, since

translation not expressed as a matrix multiplication Homogeneous coordinates allow all

three to be expressed homogeneously, using multiplication by 3  matrices W is 1 for affine transformations in

graphics

Classification of

Transformations

16

1. Rigid-body Transformation  (^) Preserves parallelism of lines  (^) Preserves angle and length  (^) e.g. any sequence of R() and T(dx,dy)
2. Affine Transformation  (^) Preserves parallelism of lines  (^) Doesn’t preserve angle and length  (^) e.g. any sequence of R(), S(sx,sy) and T(dx,dy)

unit cube (^) 45 deg rotaton Scale in X not in Y

Commutative of Transformation

Matrices

17

• (^) In general matrix multiplication is not commutative
• (^) For the following special cases commutativity holds i.e. M1.M2 = M2.M

M 1 M 2

Translate Translate Scale Scale Rotate Rotate Uniform Scale Rotate

• (^) Some non-commutative Compositions:  (^) Non-uniform scale, Rotate  (^) Translate, Scale  (^) Rotate, Translate

Original Transitional Final

Rotation of  about

P(h,k): R  ,P

19

=^ R  ,P

Q(x,y)

P(h,k)

Step 1: Translate P(h,k) to origin

T(-h ,-k)

Q1(x’,y’)

P1 (0,0)

Step 2: Rotate  w.r.t to origin

*^ R 

Q2(x’,y’)

P2 (0,0)

Step 3: Translate (0,0) to P(h,k0)

*** T(h ,k)**

P3(h,k)

Q3(x’+h, y’ +k)

Scaling w.r.t. P(h,k):

Ssx,sy,p

20

=Ssx,sy,P T(h ,k)S(sx,sy) T(-h ,-k)

S3/2,1/2,(1,1)

Step 1: Translate P(h,k) to origin

T(-1,-1)^ (

Step 2: Scale S(sx,sy) w.r.t origin

( 0 S(3/2,1/2)

Step 3: Translate (0,0) to P(h,k)

( 1 T(1,

)