Lecture Notes:

Computer Graphics

Why Transformations?

• In graphics, once we have an object described, transformations are used to move that object, scale it and rotate it

Transformation

• Two complementary views– Geometric trans: Object is transformed relative to a stationary

coordinate system

– Coordinate trans: Object is held stationary while the coordinate system is transformed

• Ex: moving an automobile against a scenic background– Move the auto and keep the background fixed (GT)

– Keep the car fixed and move the background (CT)

Basic 2D Geometric Transformations

• Four basic types– Translation

– Scaling

– Rotation

– Shear (possible using a combination of rotation and scaling)

Translation

• Simply moves an object from one position to another

xnew = xold + tx ynew = yold + ty

Note: House shifts position relative to origin

y

x 0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

Scaling

• Scalar multiplies all coordinates

• WATCH OUT: Objects grow and move!

xnew = Sx × xold ynew = Sy × yold

Note: House shifts position relative to origin

y

x 0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

1

2

1

3

3

6

3

9

Rotation

• Rotates all coordinates by a specified angle

xnew = xold × cosθ – yold × sinθ

ynew = xold × sinθ + yold × cosθ

• Points are always rotated about the origin

6

y

x 0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

Shear

• Along X-axis

• Along Y-axis

yhxxx s

xhyyy s

Basic 2D Geometric Transformations

• Translation– –

• Scale– –

• Rotation– –

• Shear

– –

txxx tyyy

sxxx syyy

sinθy-cosθxx cosθysinθxy

yhxxx sxhyyy s

Transformation and Matrix Multiplication

• Recall how matrix multiplication takes place:

ziyhxg

zfyexd

zcybxa

z

y

x

ihg

fed

cba

***

***

***

Matrix Representation

• Represent a 2D Transformation by a Matrix

• Apply the Transformation to a Point

y

x

dc

ba

y

x

dycxy

byaxx

dc

ba

TransformationMatrix Point

Matrix Representation

• Transformations can be combined by matrix multiplication

y

x

lk

ji

hg

fe

dc

ba

y

x

Matrices are a convenient and efficient way

to represent a sequence of transformations

TransformationMatrix

2×2 Matrices

• What types of transformations can be represented with a 2×2 matrix?

2D Identity

2D Scaling

yy

xx

ysyy

xsxx

y

x

y

x

10

01

y

x

sy

sx

y

x

0

0

2×2 Matrices

• What types of transformations can be represented with a 2×2 matrix?

2D Rotation

2D Shearing

y

x

y

x

cossin

sincos

y

x

shy

shx

y

x

1

1

yxy

yxx

cossin

sincos

yxshyy

yshxxx

2×2 Matrices

• What types of transformations can be represented with a 2×2 matrix?

2D Mirror over Y axis

2D Mirror over (0,0)

yy

xx

yy

xx

y

x

y

x

10

01

y

x

y

x

10

01

2×2 Matrices

• What types of transformations can be represented with a 2×2 matrix?

2D Translation

txxx tyyy

NO!!

Only linear 2D transformationscan be Represented with 2x2 matrix

Homogeneous Coordinates

• A point (x, y) can be re-written in homogeneous coordinates as (xh, yh, h)

• The homogeneous parameter h is a non-zero value such that:

• We can then write any point (x, y) as (hx, hy, h)

• We can conveniently choose h = 1 so that (x, y) becomes (x, y, 1)

h

xx h

h

yy h

Homogeneous Coordinates

• Add a 3rd coordinate to every 2D point– (x, y, w) represents a point at location (x/w, y/w)

– (x, y, 0) represents a point at infinity

– (0, 0, 0) is not allowed

1 2

1

2

x

y

(2, 1, 1) or (4, 2, 2) or (6, 3, 3)

Convenient Coordinate System to

Represent Many Useful Transformations

Why Homogeneous Coordinates?

• Mathematicians commonly use homogeneous coordinates as they allow scaling factors to be removed from equations

• We will see in a moment that all of the transformations we discussed previously can be represented as 3*3 matrices

• Using homogeneous coordinates allows us use matrix multiplication to calculate transformations – extremely efficient!

Homogeneous Translation

•The translation of a point by (dx, dy) can be written in matrix form as:

• Representing the point as a homogeneous column vector we perform the calculation as:

100

10

01

dy

dx

11*1*0*0

1**1*0

1**0*1

1100

10

01

dyy

dxx

yx

dyyx

dxyx

y

x

dy

dx

Homogenous Coordinates

• To make operations easier, 2-D points are written as homogenous coordinate column vectors

vdydxTvdyy

dxx

y

x

dy

dx

),(':

11100

10

01

vssSvys

xs

y

x

s

s

yxy

x

y

x

),(':

11100

00

00

Translation:

Scaling:

Basic 2D Transformations

• Basic 2D transformations as 3x3 Matrices

1100

10

01

1

y

x

ty

tx

y

x

1100

0cossin

0sincos

1

y

x

y

x

1100

00

00

1

y

x

sy

sx

y

x

1100

01

01

1

y

x

shy

shx

y

x

Translate

Shear

Scale

Rotate

Combining Transformations

•A number of transformations can be combined into one matrix to make things easy

– Allowed by the fact that we use homogenous coordinates

• Imagine rotating a polygon around a point other than the origin

– Transform to centre point to origin

– Rotate around origin

– Transform back to centre point

Matrix Composition

• Transformations can be combined by matrix multiplication

• Efficiency with premultiplication

– Matrix multiplication is associative

w

y

x

1 0 0

0sy0

00sx

1 0 0

0cosθinθ

0 sinθ- θcos

100

10

01

sty

tx

w

y

x

p ty)T(tx, )R( sy)S(sx, p

p)))S(R(T(p pS)RT(p

Combining Transformations (cont…))(HHouse HdydxT ),(

HdydxTR ),()( HdydxTRdydxT ),()(),(

1 2

3 4

Combining Transformations (cont…)

• The three transformation matrices are combined as follows

1100

10

01

100

0cossin

0sincos

100

10

01

y

x

dy

dx

dy

dx

REMEMBER: Matrix multiplication is not commutative so order matters

vdydxTRdydxTv ),()(),('

Matrix Composition

• Rotate by around arbitrary point (a,b)

–

• Scale by sx, sy around arbitrary point (a,b)

–

a,-b-TRba,TM θ

a,-b-Tsysx,Sba,TM

(a,b)

(a,b)

Inverse Transformations

•Transformations can easily be reversed using inverse transformations

100

10

011 ty

tx

T

100

01

0

001

1

y

x

s

s

S

100

0cossin

0sincos1

R

Basic 2D Coordinate Transformations

• Translation

–

–

• Scale

–

–

• Rotation

–

–

• Shear ??

txxx tyyy

sx)/1(xx

sy)/1(yy

sinθycosθxx cosθysinθxy