April 2010 1

Geometric Transformations forComputer Graphics

Shmuel Wimer

Bar Ilan Univ., School of Engineering

April 2010 2

2D Translation

P

P

T

x

y ,

, ,

x y

x

y

x x t y y t

tx x

ty y

P P T

P P T

April 2010 3

2D Rotation

x

y

rx

ry

Rotation in angle about a

pivot (rotation) point , .r rx y

x

y

,r rx y

,x y

,x y

cos sin

sin cos

cos sin

sin cos

r r r

r r r

r r

x x x x y y

y y x x y y

P P R P P

R

April 2010 4

2D Scaling

xS

yS

,

0

0

x y

x

y

x x s y y s

sx x

sy y

P S P

x

y

,f fx y

1

1

x f x

y f y

f

x x s x s

y y s y s

P P S P 1- S

Scaling about a fixed point ,f fx y

April 2010 5

Homogeneous Coordinates

1 2

1 2

Rotate and then displace a point :

: 2 2 rotation matrix. : 2 1 displacement vector.

P P = M P + M

M M

Displacement is unfortunately a non linear operation.

Make displacement linear with .

, , ,1 . Transformations turn into 3 3 matrices.x y x y

Homoheneous Coordinates

Very big advantage. All transformations are concatenated by

matrix multiplication.

April 2010 6

1 0

0 1 , ,

1 0 0 1 1

x

y x y

x t x

y t y t t

P T P2D Translation

cos sin 0

sin cos 0 ,

1 0 0 1 1

x x

y y

P R P2D Rotation

0 0

0 0 , ,

1 0 0 1 1

x

y x y

x S x

y S y S S

P S P2D Scaling

April 2010 7

1 1 1

1 0 cos sin 0 1 0 0

0 1 , sin cos 0 , 0 1 0

0 0 1 0 0 1 0 0 1

x x

y y

t S

t S

T R S

Inverse transformations:

2 1 2 1 P M M P M M P M PComposite transformations:

2 2 1 1 2 2 1 1

2 1 1 2

2 1 1 2

2 2 1 1 1 2 1 2

, , , ,

1 0 1 0 1 0

0 1 0 1 0 1

0 0 1 0 0 1 0 0 1

, , ,

x y x y x y x y

x x x x

y y y y

x y x y x x y y

t t t t t t t t

t t t t

t t t t

t t t t t t t t

P T T P T T P

T T T

Composite translations:

April 2010 8

2 1 2 1

2 1 1 2

1 2

P R R P R R P

R R R

P R P

Composite Rotations:

2 1 1 2

2 1 1 2

2 2 1 1 1 2 1 2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 1 0 0 1 0 0 1

, , ,

x x x x

y y y y

x y x y x x y y

S S S S

S S S S

S S S S S S S S

S S SComposite Scaling:

April 2010 9

General 2D Rotation

Move to origin Rotate Move back

1 0 cos sin 0 1 0

0 1 sin cos 0 0 1

0 0 1 0 0 1 0 0 1

cos sin 1 cos sin

sin cos 1 cos sin

0 0 1

r r

r r

r r

r r

x x

y y

x y

y x

,r rx y

April 2010 10

General 2D Scaling

Move to origin Scale Move back

0 11 0 0 0 1 0

0 1 0 0 0 1 0 1

0 0 1 0 0 1 0 0 1 0 0 1

x f xf x f

f y f y f y

S x Sx S x

y S y S y S

,f fx y

April 2010 11

2D Directional Scaling

y

x2S

1S

11 2

2 21 2 2 1

2 22 1 1 2

,

cos sin cos sin 0

cos sin sin cos 0

0 0 1

S S

S S S S

S S S S

R S R

y

x

y

x

1

2

O

1

2

45

S

S

April 2010 12

2D Reflections

x

y

x

y

x

y

1 0 0

0 1 0

0 0 1

1 0 0

0 1 0

0 0 1

1 0 0

0 1 0

0 0 1

April 2010 13

0 1 0

1 0 0

0 0 1

3

1

2

1

2

3

y x y x

12

3

1

2

3

0 1 0

1 0 0

0 0 1

April 2010 14

Geometric Transformations by Rasterization

• The transformed shape needs to be filled.– A whole scan-line filling is usually in order.

• However, simple transformations can save new filling by manipulating blocks in the frame buffer.

minP

maxP

0P

Translation:

Move block of pixels of frame buffer into new destination.

April 2010 15

1 2 3

4 5 6

7 8 9

10 11 12

3 6 9 12

2 5 8 11

1 4 7 10

90° counterclockwise rotation

12 11 10

9 8 7

6 5 4

3 2 1

180° rotation

Rotated pixel block

Destination pixel array

RGB of destination pixel can be determined by averaging rotated ones (as antialiasing)

April 2010 16

Translation

, ,x y z

, ,x y z

x

y

z

x

y

z

x x t

y y t

z z t

3D Transformations

1 0 0

0 1 0

0 0 1

1 0 0 0 1 1

x

y

z

x t x

y t y

z t z

Very similar to 2D. Using 4x4 matrices rather than 3x3.

April 2010 17

General 3D Rotation

1. Translate the object such that rotation axis passes

through the origin.

2. Rotate the object such that rotation axis coincides with

one of Cartesian axes.

3. Perform specified rotation about the Cartesian axis.

4. Apply inverse rotation to return rotation axis to original

direction.

5. Apply inverse translation to return rotation axis to

original position.

April 2010 18

x

z

y2P

1P

x

z

y

2P

1Px

z

y

1P

2P

x

z

y

1P

2P x

z

y

2P

1Px

z

y2P

1P

April 2010 19

1 2

2 1 2 1 2 1 2 1

The vector from to is:

, ,x x y y z z

P P

V P P

2 1

2 1

2 1

2 2 2

Unit rotation vector: | | , ,

| |

| |

| |

1

a b c

a x x

b y y

c z z

a b c

u V V

V

V

V

1

1

1

1 0 0

0 1 0

0 0 1

0 0 0 1

x

y

z

T

, ,a b cu

x

y

z

April 2010 20

Rotating to coincide with axiszu

2 2

2 2

First rotate around axis to lay in plane.

Equivqlent to rotation 's projection on plane around axis.

cos , sin .

We obtained a unit vector ,0, in plane.

x x z

y z x

c b c c d b d

a b c d x z

u

u

w

1 0 0 0

0 0

0 0

0 0 0 1

x

c d b d

b d c d

R

, ,a b cu

x

y

z

u

April 2010 21

, ,a b cu

x

y

z

,0,a dw

2 2

Rotate counterclockwise around axis.

is a unit vector whose component is , component is 0,

hence component is .

y

x a y

z b c d

w

w

cos , sind a

0 0

0 1 0 0

0 0

0 0 0 1

y

d a

a d

R

1 1 1x y z y x R T R R R R R T

April 2010 22

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

z

R

1 1 1x y z y x R T R R R R R T

2

2

2

1 cos cos 1 cos sin 1 cos sin

1 cos sin 1 cos cos 1 cos sin

1 cos sin 1 cos sin 1 cos cos

R

a ab c ac b

ba c b bc a

ca b cb a c

M

April 2010 23

Efficient 3D Rotations by Quaternions

Quaternions are extensions of complex numbers to 4-dimension.

, : real , , : imaginaryq s ia jb kc s s a b c v v

2 2 2 1, , , i j k ij ji k jk kj i ki ik j

1 2 1 2

1 2 1 2 1 2 2 1

Addition: ,

Multiplication: ,

q q s s

q q s s s s

1 2

1 2 1 2

v v

v v v + v v v

2 2 1 1 12

1, , , 1,0q s q s qq q q

q v v v

April 2010 24

x

z

y

u

Rotate a point position , , about the unit vector .x y zp u

Quarternion representation:

Rotation: cos , sin2 2

Position: 0, , , ,

q

x y z

u

P p p

1 2

Rotation of is carried out with the quarternion operation:

0, 2

This can be calculated by efficient HW for fast 3D rotations

when many rotation operations are involved.

q q s s

P

P P p v p v v p v v p

April 2010 25

3D Scaling

x

y

z x

y

z

x

y

z

x x S

y y S

z x S

Enlarging object also moves it from origin

0 0 0

0 0 0

0 0 0

1 0 0 0 1 1

x

y

z

x S x

y S y

z S z

P S P

April 2010 26

, ,f f fx y z

x

y

z

Scaling with respect to a fixed point (not necessarily of object)

, ,f f fx y z

x

y

z , ,f f fx y z

x

y

z

, ,f f fx y z

x

y

z

1

0 0 1

0 0 1

0 0 1

0 0 0 1

x x f

y y f

z z f

S S x

S S y

S S z

T S T