CG Lect 11 2DTransformations

8/2/2019 CG Lect 11 2DTransformations

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Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi

Lecture 11: 2D Transformations CS 433: Computer Graphics

2D Transformations

Dr. Zahid Halim


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DefinitionsDefinitionsA matrix is a rectangular array of elements (usuallynumbers) arranged in rows and columns enclosed inbrackets.

=

=

=

d

c

b

a

C B A

213

532042

20

4121

Order of the matrix is the number of rows and columns. A is a 3 2matrix, B is a 3 3 matrix C is a 4 1 matrix .Matrices are denoted by capital letters while their elements are writtenas lower case letters as in example C above.

We refer to a particular element by using notation that refers to the rowand column containing the element, i.e. a 21=-1, b 32 = 1, c 21 = b.Two matrices A and B are said to be equal, written A = B, if they havethe same order and a ij = b ij for every i and j .


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Type of MatricesType of MatricesSquare matrix a matrix with an equal number ofrows and columns.

Diagonal matrix a square matrix with zeros

everywhere except down the leading diagonal.

Unit matrix (identity matrix ) a diagonal matrix withones down the leading diagonal. The identity matrixis denoted with the letter I .

matrixunit33ais

100

010001

3

= I

matrixdiagonal33ais 900050001

= A

matrixsquare33ais 987654321

= A


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Type of Matrices (continue)Type of Matrices (continue)Zero matrix a matrix with zeros everywhere,denoted by O .

Symmetric matrix a square matrix whose ( ij )th

element is the same as the ( ji )th element for all i and j .

Row matrix (row vector ) a matrix with only one row,i.e. its order is 1 m .Column matrix (column vector ) a matrix with onlyone column, i.e. an n 1 matrix.

matrixsymmetricais 374725451

= A


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Matrix Addition and SubtractionMatrix Addition and SubtractionIf A and B are two n m matrices then their sum C = A + B is the n m matrix with

c ij = a ij + b ij

their difference C = A - B is the n m matrix with

c ij = a ij - b ij

=

=

1024

172

7)3()3()1()2(2

432)5(31

732

423

312

351

=

+++

+++=

+

440

734

7)3()3()1()2(2

432)5(31

732

423

312

351

Examples:Examples:


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Scalar Multiplication of MatricesScalar Multiplication of Matrices

If A is an n m matrix and k is a real number,then the scalar multiple C = kA is the n m matrix where c ij = ka ij .

=

=

1464

846

72)3(2)2(2

422232

732

4232


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The Transpose of a MatrixThe Transpose of a Matrix

If A is a matrix then the transpose of A is the

matrix At where a

t ij = a ji .

=

312412023121

A

=

340111222231

t A


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Matrix MultiplicationMatrix Multiplication

If A is an n m matrix and B is an m p matrixthen the product of A and B , C = AB , is ann p matrix

C ij = (row i of A) (column j of B )

= a i1b 1j + a i2 b 2j + a i3 b 3j + a in b jp

232333

45

214

83

02)2(041)3(23011

01)2()3(4)2()3(13)3(1)2(

04)2(243)3(43213

03

23

41

201

132

423

=

++++

++++

++++=


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PropertiesProperties

If a , b are scalars and A, B , C are matrices ofthe same dimension (order) then

A + B = B + AA + (B + C) = (A + B) + C a(bA) = (ab)A(a + b)A = aA + bAa(A + B) = aA + aB a(A - B) = aA - aB A + OA = A, and aO = O (A + B) t = At + B t

(A - B) t = At - B t


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PropertiesProperties

If a , b are scalars and A, B , C are matricesof the same dimension (order) then

assuming that the matrix dimensions aresuch that the products in each of thefollowing are defined, then we have:

AB BAA(BC) = (AB)C (A + B)C = AC + BC A(B + C) = AB + AC OA = O, AO = O

(aA)(bB) = abAB IA = AI = A(AB)T = B T AT


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Inverse of a 2Inverse of a 2

2 Matrix2 Matrix

Not all Matrices have an inverse.The Determinant determines if a matrix has an

inverse or not. If the determinant of a matrix A is zerothen the matrix has no inverse.

bcad A Ad c ba A==

= det

11)1(342det 4132

===

= A A A

Example:Example:


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Inverse of a 2Inverse of a 2 2 Matrix2 Matrix

If A is a matrix then it is possible to find anothermatrix, called A-1, such that AA-1 = I or A-1A = I .

=

=

acbd

A A

d cba

A1

1

=

=

=

112

111

113

114

1

21 34111 41 32 A A

Example:Example:


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Solution of Systems ofSolution of Systems ofSimultaneous EquationsSimultaneous Equations

4

1132=+

=+

y x

y x

=

411

1132

y x

AA x x == b b

Ax = b A-1Ax = A-1b x = A-1b


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=

41111 32 y x

AA x x == b b

31

31

31

4)2(1114311)1(

411

2131

2131

2131

1132

1

111

=

=

=

=

=

+

+=

==

=

=

=

y x

y x

x

b A x

A A


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Inverse of a 3Inverse of a 3 3 matrix3 matrix

++

+

++

=

======

======

======

++++==

=

=

1

:tdeterminantheFrom

)()(det

333231

232221

1312111

333231

232221

131211

aaaaaa

aaa

A A

bd aeeb

d aabgah

hb

gaaegdh

he

gd a

cd af f cd a

acgaiicga

a fgdii f gd

a

cebf f ceb

achbiichb

a fheii f he

a

A

afhbdicegcdhbfgaei A A

i f chebgd a

Aihg

f ed cba

A

t

t


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Inverse of a 3 3 matrix (example)

++

+

++

=

=

====

=

=

====

=

======

==+++==

=

=

444433437

41

431

114

41

014

43

01

44011

33001

33401

4

40

313

30

417

34

43

:tdeterminantheFrom

4139)3160()009(det

340431011

340431011

1

333231

232221

131211

A

aaa

aaa

aaa

A

A A

A A

t

t


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Why Transformations? In graphics, once we have an object described, transformations

are used to move that object, scale it and rotate it


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Translation Simply moves an object from one position to another

xnew = x old + dx y new = y old + dy

Note: House shifts position relative to origin

y

x0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6


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Translation Example y

x0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(1, 1) (3, 1)

(2, 3)


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Scaling Scalar multiplies all coordinates WATCH OUT: Objects grow and move! xnew = Sx x old ynew = Sy y old

Note: House shifts position relative to origin

y

x0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

12

13

3

6

3

9


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Scaling Example y

0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(1, 1) (3, 1)

(2, 3)


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Rotation Rotates all coordinates by a specified angle xnew = xold cos yold sin

ynew = xold sin + yold cos Points are always rotated about the origin

6

=

y

x01

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6


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Rotation Example y

0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(3, 1) (5, 1)

(4, 3)


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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 x x h=

h y y h=


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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 wediscussed previously can be represented as 3*3 matrices

Using homogeneous coordinates allows us use matrixmultiplication to calculate transformations extremely efficient!


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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 weperform the calculation as:

100

1001

dydx

+

+

=

++

++

++

=

11*1*0*0

1**1*0

1**0*1

1100

10

01

dy y

dx x

y x

dy y x

dx y x

y

x

dy

dx


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Remember Matrix Multiplication Recall how matrix multiplication takes place:

++

++

++

=

zi yh xg

z f ye xd zc yb xa

z

y x

ihg

f ed cba

***

******


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Homogenous Coordinates To make operations easier, 2-D points are written as

homogenous coordinate column vectors

vdydxT vdy y

dx x

y

x

dy

dx

),(':

11100

10

01=+

+

=

vssSv ys

xs

y

x

s

s

y x y

x

y

x

),(':

11100

00

00=

=

Translation:

Scaling:


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Homogenous Coordinates (cont)

v Rv y x

y x

y

x

)(':

1

cossin

sincos

1100

0cossin

0sincos

=+

=

Rotation:


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Inverse Transformations Transformations can easily be reversed using inverse

transformations

=

100

10

011 dy

dx

T

=

100

01

0

001

1

y

x

s

s

S

=

100

0cossin

0sincos1

R


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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

f h


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Combining Transformations (cont))( H House H dydxT ),(

H dydxT R ),()( H dydxT RdydxT ),()(),(

1 2

3 4

L 11 2D T f i CS 433 C G hi


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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 notcommutative so order matters

vdydxT RdydxT v ),()(),(' =

L t 11 2D T f ti CS 433 C t G hi


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Reference Book Code: A

Chapter: 5

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CG Lect 11 2DTransformations