IT- 601: Computer GraphicsLecture-06

Two –DimensionalTransformations

Jesmin Akhter Lecturer,IIT

Jahangirnagar University, Savar, Dhaka,Bangladesh

04/22/23

Matrix Math• Why do we use matrix?

– More convenient organization of data.– More efficient processing– Enable the combination of various

concatenations

• Matrix addition and subtraction

ab

cd

a cb d=

• Matrix Multiplication– Dot product

Matrix Math

a bc d

e fg h

. = a.e + b.g a.f + b.hc.e + d.g c.f + d.h

• What about this?

• Type of matrix

Matrix Math

1 2 . 1 23 1

= 7 4

23

. 1 23 1 = No!!

a b ab

Row-vector Column-vector

• Is there a difference between possible representations?

Matrix Math

dfce

bfae

f

e

dc

ba

dfbecfae dc

bafe

dfcebfae db

cafe

• The column-vector representation for a point.

DyCx

ByAx

y

x

DC

BA

Matrix Math

• Example– Find the transformed point, P’, caused by

rotating P= (5, 1) about the origin through an angle of 90.

Rotation

cossin

sincos

cossin

sincos

yx

yx

y

x

90cos190sin5

90sin190cos5

0115

1105

5

1

Composite Transformation• We can represent any sequence of transformations

as a single matrix.– Composite transformations – matrix • matrix.

• Composite transformations:– Rotate about an arbitrary point – translate, rotate,

translate– Scale about an arbitrary point – translate, scale,

translate– Change coordinate systems – translate, rotate, scale

• Does the order of operations matter?

Composition Properties

• Is matrix multiplication associative?– (A.B).C = A.(B.C)

dhlcfldgjcejdhkcfkdgicei

bhlaflbgjaejbhkafkbgiaei

lk

ji

dhcfdgce

bhafbgae

lk

ji

hg

fe

dc

ba

dhldgjcflcejdhkdgicfkcei

bhlbgjaflaejbhkbgiafkaei

hlgjhkgi

flejfkei

dc

ba

lk

ji

hg

fe

dc

ba

?

• Is matrix multiplication commutative?– A . B = B . A

Composition Properties

?

dhcfdgce

bhafbgae

hg

fe

dc

ba

hdgbhcga

fdebfcea

dc

ba

hg

fe

Order of operationsSo, it does matter. Let’s look at an example:

1. Translate2. Rotate

1. Rotate

2. Translate

Composite Transformation Matrix• Arrange the transformation matrices in order from right to left.

• General Pivot- Point Rotation• Operation :-

1. Translate (pivot point is moved to origin)2. Rotate about origin

3. Translate (pivot point is returned to original position)

T(pivot) • R() • T(–pivot)

1 0 -tx

0 1 -ty

0 0 1

cos -sin 0sin cos 0 0 0 1

1 0 tx

0 1 ty

0 0 1 . .

cos -sin -tx cos+ ty sin + tx

sin cos -tx sin - ty cos + ty

0 0 1

cos -sin -tx cos+ ty sin sin cos -tx sin - ty cos 0 0 1

1 0 tx

0 1 ty

0 0 1 .

• Example– Perform 60 rotation of a point P(2, 5) about a

pivot point (1,2). Find P’?

Composite Transformation Matrix

cos -sin -tx cos+ ty sin + tx

sin cos -tx sin - ty cos + ty

0 0 1

xy1

.

0.5 -0.866 -1.0.5 + 2.0.866 + 1 0.866 0.5 -1.0.866- 2.0.5 + 20 0 1

251

.

0.5 - 0.866 2.232 0.866 0.5 0.1340 0 1

251

. =-1.0984.366 1

P’ = (-1, 4)

Sin 60 = 0.8660Cos 60 = 1/2

Composite Transformation Matrix

General Fixed-Point Scaling Operation :-

1. Translate (fixed point is moved to origin)2. Scale with respect to origin

3. Translate (fixed point is returned to original position)

T(fixed) T(fixed) • • S(scale) S(scale) • • T(–fixed)T(–fixed)

Find the matrix that represents scaling of an object with respect to any fixed point?

Given P(6, 8) , Sx = 2, Sy = 3 and fixed point (2, 2). Use that matrix to find P’?

Answer1 0 -tx

0 1 -ty

0 0 1

Sx 0 0 0 Sy 0 0 0 1

1 0 tx

0 1 ty

0 0 1 . .

Sx 0 -tx Sx 0 Sy -ty Sy 0 0 1

1 0 tx

0 1 ty

0 0 1 . =

x =6, y = 8, Sx = 2, Sy = 3, tx =2, ty = 2

Sx 0 -tx Sx + tx

0 Sy -ty Sy + ty

0 0 1

2 0 -2( 2) + 20 3 -2(3) + 20 0 1

.681

=10 20 1

Composite Transformation Matrix

General Scaling Direction Operation :-

1. Rotate (scaling direction align with the coordinate axes)2. Scale with respect to origin

3. Rotate (scaling direction is returned to original position)

R(–) • S(scale) • R()

Find the composite transformation matrixby yourself !!

Other transformationsReflection:

x-axis y-axis

100

010

001

100

010

001

Other transformationsReflection:

origin

100

010

001

Other transformationsShear:

x-direction y-direction

100

010

01 xsh

100

01

001

ysh

Another Example.

Scale

Translate

Rotate

Translate