1

2D Geometric Transformations

(Chapter 5 in FVD)

2

2D geometric transformation

• Translation

• Scaling

• Rotation

• Shear

• Matrix notation

• Compositions

• Homogeneous coordinates

3

2D Geometric

Transformations

• Question: How do we represent a

geometric object in the plane?

• Answer: For now, assume that

objects consist of points and lines.

A point is represented by its

Cartesian coordinates: (x,y).

• Question: How do we transform a

geometric object in the plane?

• Answer: Let (A,B) be a straight line

segment and T a general 2D

transformation: T transforms (A,B)

into another straight line segment

(A’,B’), where A’=TA and B’=TB.

4

Translation

• Translate (a,b): (x,y) (x+a,y+b)

Translate(2,4)

5

Scale

• Scale (a,b): (x,y) (ax,by)

Scale (2,3)

Scale (2,3)

6

• How can we scale an object

without moving its origin (lower

left corner)?

Translate(-1,-1)

Translate(1,1) Scale(2,3)

7

Reflection

Scale(-1,1)

Scale(1,-1)

8

Rotation

• Rotate(q):

(x,y) (x cos(q)+y sin(q), -x sin(q)+y cos(q))

Rotate(90)

Rotate(90)

9

• How can we rotate an object

without moving its origin (lower

left corner)?

Translate(-1,-1)

Translate(1,1)

Rotate(90)

10

Shear

• Shear (a,b): (x,y) (x+ay,y+bx)

Shear(1,0)

Shear(0,2)

11

Composition of

Transformations

• Rigid transformation:

– Translation + Rotation (distance

preserving).

• Similarity transformation:

– Translation + Rotation + uniform

Scale (angle preserving).

• Affine transformation:

– Translation + Rotation + Scale +

Shear (parallelism preserving).

• All above transformations are groups

where Rigid Similarity Affine.

12

Rigid

Affine

Similarity

13

Matrix Notation

• Let’s treat a point (x,y) as a 2x1

matrix (a column vector):

• What happens when this vector is

multiplied by a 2x2 matrix?

y

x

dycx

byax

y

x

dc

ba

14

2D Transformations

• 2D object is represented by points

and lines that join them.

• Transformations can be applied

only to the the points defining the

lines.

• A point (x,y) is represented by a

2x1 column vector, and we can

represent 2D transformations

using 2x2 matrices:

y

x

dc

ba

y

x

'

'

15

Scale

• Scale(a,b): (x,y) (ax,by)

• If a or b are negative, we get

reflection.

• Inverse: S-1(a,b)=S(1/a,1/b)

by

ax

y

x

b

a

0

0

16

Reflection

• Reflection through the y axis:

• Reflection through the x axis:

• Reflection through y=x:

• Reflection through y=-x:

10

01

10

01

01

10

01

10

17

Shear, Rotation

• Shear(a,b): (x,y) (x+ay,y+bx)

• Rotate(q):

(x,y) (xcosq+ysinq , -xsinq + ycosq)

• Inverse: R-1(q)=RT(q)=R(-q)

bxy

ayx

y

x

b

a

1

1

qq

qq

qq

qq

cossin

sincos

cossin

sincos

yx

yx

y

x

18

Composition of Transformations

• A sequence of transformations

can be collapsed into a single

matrix:

• Note: order of transformations is

important! (otherwise - commutative groups)

y

xD

y

xCBA

translate rotate

rotate translate

19

Composition of Transformations

(Cont.)

D = A B C

D-1 = C-1B-1A-1

D * D-1 = ABC * C-1B-1A-1 =

AB*I*B-1*A-1=

A*I*A = I

Proof:

20

Translation

• Translation(a,b):

• Problem: Cannot represent

translation using 2x2 matrices.

• Solution:

Homogeneous Coordinates

by

ax

y

x

21

Homogeneous Coordinates

• Homogeneous Coordinates is a

mapping from Rn to Rn+1:

• Note: (tx,ty,t) all correspond to

the same non-homogeneous point

(x,y). E.g. (2,3,1)(6,9,3).

• Inverse mapping:

),,(),,(),( ttytxWYXyx

W

Y

W

XWYX ,),,(

22

Translation

• Translate(a,b):

• Inverse: T-1(a,b)=T(-a,-b)

• Affine transformation now have

the following form:

11100

10

01

by

ax

y

x

b

a

100

fdc

eba

23

(X,Y,W)

y

x

X

Y

W

1

(X,Y,1)

Geometric Interpretation

• A 2D point is mapped to a line

(ray) in 3D. The non-homogeneous

points are obtained by projecting

the rays onto the plane Z=1.

24

• Example: Rotation about an

arbitrary point:

• Actions:

– Translate the coordinates so that the

origin is at (x0,y0).

– Rotate by q.

– Translate back.

(x0,y0)

q

1100

sin)cos1(cossin

sin)cos1(sincos

1100

10

01

100

0cossin

0sincos

100

10

01

00

00

0

0

0

0

y

x

xy

yx

y

x

y

x

y

x

qqqq

qqqq

qq

qq

25

• Another example: Reflection

about an Arbitrary Line:

• Actions:

– Translate the coordinates so that P1

is at the origin.

– Rotate so that L aligns with the x-

axis.

– Reflect about the x-axis.

– Rotate back.

– Translate back.

p1

p2

L=p1+t (p2-p1)=t p2+(1-t) p1

26

Change of Coordinates

• It is often requires the transformation of

object description from one coordinate

system to another.

• How do we transform between two

Cartesian coordinate systems?

• Rule: Transform one coordinate frames

towards the other in the opposite

direction of the representation change.

x

y

x’y’

27

Change of Coordinates (Cont.)

X

Y

P’

O

P

X’

Y’O’

Example:

28

• Example:

– Represent the point P=(xp,yp,1) in the

(x’,y’) coordinate system.

x

y x’y’

q

(x0,y0)(xp,yp)

MPP '

100

10

01

100

0cossin

0sincos

0

0

11 y

x

TRM qq

qq

where

29

• Alternative method:

– Assume x’=(ux,uy) and y’=(vx,vy) in the

(x,y) coordinate system .

x

yx’y’

(x0,y0)

MPP '

100

10

01

100

0

0

0

0

y

x

vv

uu

M yx

yx

where

(ux,uy)(vx,vy)

30

• Example:

– P is at the y’ axis P=(vx,vy):

– What is the inverse?

x

y

x’y’

(ux,uy)

(vx,vy)

110

1100

0

0

' y

x

yx

yx

v

v

vv

uu

MPP

31

• Another example: Reflection about an

Arbitrary Line:

• Define a coordinate systems (u,v)

parallel to P1P2:

p1

p2

x

y

y

x

u

u

pp

pp

12

12u

y

x

x

y

v

v

u

uv

100

10

01

100

0

0

100010001

100

0

0

100

10

01

1

1

1

1

y

x

y

x

p

p

vv

uu

vu

vu

p

p

M yx

yx

yy

xx