7. Transformation- 2D

8/3/2019 7. Transformation- 2D

1/93

2D- Transformation


2/93

2D Transformations

Transformations

The geometrical changes of an object from a current state tomodified state.

Includes changing the size of an object, position on the screen and

orientation.

Transformations is needed

To manipulate the initially created object and to display the

modified object without having to redraw it.

Transformed Matrix = [ Original Object Matrix] * [ T/f Matrix ].


3/93

Two Ways

Object Transformation :-

Alter the coordinates descriptions an object

Translation, rotation, scaling etc.

Object is moved relative to Coordinate System

Coordinate system unchanged

Coordinate Transformation :-

Object held stationary. Coord System is moved relative to the Object.

Produce a different coordinate system

2D Transformations


4/93

Geometric Transformation :

Operations that are applied to the geometric description of an

object to change its position, orientation, or size are called

Geometric Transformations.

In this object is moved relative to stationary object, coordinatesystem or background.


5/93

So the geometric-transformation functions that are available in somesystem are following:

1. Translation

2. Rotation

3. Scaling

4. Other useful transforms includes: reflection and shear.


6/93

Translation

Rotation


7/93

Scaling

Uniform Scaling

Un-uniform Scaling


8/93

Reflection

Shear


9/93

Why do we need geometric transformations in

CG( motivation )?

As a viewing aid

As a modeling tool

As an image manipulation tool


10/93


11/93

Two-Dimensional(2D) Translation

Suppose tx and ty is the translation distances, (x, y) is the original

coordinates, is the new coordinate position.

txxx

y y ty

'

( , ')x y

x x tx

y y ty


12/93

Re-position a point along a straight line

Given a point (x,y), and the translation

distance (tx,ty)The new point: (x, y)

x = x + txy = y + ty

(x,y)

(x,y)

OR P = P + T where P = x p = x T = txy y ty

tx

ty


13/93

3x3- 2D Translation Matrix

x = x + txy y ty

Use 3 x 1 vector

x 1 0 tx xy = 0 1 ty * y1 0 0 1 1

Note that now it becomes a matrix-vector multiplication


14/93


15/93

r is the constant distance of the point formthe origin, angle is the original angular

position of the point from the horizontal,

and is the rotation angle.

Substitute expression (1) into (2)

cos

sin

x r

y r

cossinsincos)sin('

sinsincoscos)cos('

rrry

rrrx

1

2


16/93

Matrix form

cossin'

sincos'

yxy

yxx3

cos sin

sin cosx y x y


17/93

2 D - Rotation A rotation repositions

all points in an object

along a circular path in

the plane centered at

the pivot point.

First, well assume the

pivot is at the origin.

P

P


18/93

2 D - Rotation

(x,y)

(x,y)

r

x = r cos () y = r sin ()

x = r cos ( ) y = r sin ( )

x = r cos ( )= r cos() cos() r sin() sin()

= x cos() y sin()

y = r sin ( )= r sin() cos() + r cos()sin()

= y cos() + x sin()


19/93

2 D - Rotation

(x,y)

(x,y)

(x,y) -> Rotate about the originby

(x, y)

How to compute (x, y) ?

x = r cos () y = r sin ()

r

x = r cos ( ) y = r sin ( )


20/93

2 D - Rotation

(x,y)

(x,y)

r

x = x cos() y sin()

y = y cos() + x sin()

Matrix form?

x cos() -sin() xy sin() cos() y=

3 x 3?


21/93

3x3 2D Rotation Matrix

x cos() -sin() xy sin() cos() y

=

(x,y)

(x,y)

r

x cos() -sin() 0 xy sin() cos() 0 y1 0 0 1 1

=


22/93

2 D - Rotation Review Trigonometry

=> cos = x/r , sin = y/r

x = r. cos, y = r.sin

P(x,y)

x

yr

x

y

P(x, y)

r=> cos (+ ) = x/rx = r. cos (+ )x = r.coscos -r.sinsin

x = x.cos y.sin

=>sin (+ ) = y/r

y = r. sin (+ )

y = r.cossin + r.sincos

y = x.sin + y.cos

Identity of Trigonometry


23/93

2 D - Rotation We can write the components:

p'x= pxcos pysin

p'y= pxsin + pycos

or in matrix form:

P' = R P

can be clockwise (-ve) orcounterclockwise (+ve as ourexample).

Rotation matrix

P(x,y)

x

yr

x

y

P(x, y)

cossinsincosR


24/93

2 D - Rotation Default rotation center: Origin (0,0)

can be clockwise (-ve) or counterclockwise (+ve )asour example).

> 0 : Rotate counter clockwise

< 0 : Rotate clockwise


25/93

Example Find the transformed point, P, caused by rotating P= (5,

1) about the origin through an angle of 90.

2 D - Rotation

cossin

sincos

cossin

sincos

yx

yx

y

x

90cos190sin5

90sin190cos5

0115

1105

5

1

Q estion 1


26/93

Question 1:

If the rotation point is an arbitrary pivot position ,can you give the correct rotation expression?

),(rr

yx

' ( ) cos ( )sin

' ( ) sin ( ) cos

r r r

r r r

x x x x y y

y y x x y y

3


27/93


28/93


29/93

Rotation ( Revision) Applied to an Object by repositioning it along the

circular Path.

Point can be rotated through an Anlge Q.

Q +ve : Counterclockwise Rotation [ cos QsinQ, sin Q, cos Q]

Qve: Anticlockwise Rotation. [ cos Q sin Q, -sin Q cos Q]


30/93

2D- Scaling Transformation that change the Size of or Shape of an

Object.

Scaling Factor Sx or Sy.

Scalinf Factor wrt origin is carried out by multiplying the

coordinate values (x,y) of each vertex of Polygon by Sx, Syto produce (x,y).


31/93

2D Scaling

Scale: Alter the size of an object by a scaling factor(Sx, Sy), i.e.

x = x . Sxy = y . Sy

x Sx 0 xy 0 Sy y

=

(1,1)

(2,2) Sx = 2, Sy = 2

(2,2)

(4,4)


32/93

2D Scaling

(1,1)

(2,2) Sx = 2, Sy = 2

(2,2)

(4,4)

Not only the object size is changed, it also moved!! Usually this is an undesirable effect We will discuss later (soon) how to fix it


33/93

After applying Scaling factor : Object is either Inc. or Dec.

If Greater then 1 : Enlargement Object.

Less then 1 : Compression will occur.

Uniform Scaling : Sx = Sy >1 and Object becomes Larger.

Uniform Scaling : Sx = Sy 1: Object moves far from Origin

Differential Scaling : It is non uniform comprn which depends

upon Sx and Sy < or > 1, but unequal.


34/93


35/93

3x3 2D Scaling Matrix

x Sx 0 xy 0 Sy y

=

x Sx 0 0 x

y = 0 Sy 0 * y1 0 0 1 1


36/93

Or, 3x3 Matrix representations Translation:

Rotation:

Scaling:

Why use 3x3 matrices?

x 1 0 tx xy = 0 1 ty * y1 0 0 1 1

x cos() -sin() 0 xy sin() cos() 0 * y1 0 0 1 1

=

x Sx 0 0 x

y = 0 Sy 0 * y1 0 0 1 1


37/93

Why use 3x3 matrices?

So that we can perform all transformationsusing matrix/vector multiplications

This allows us to pre-multiplyall the matricestogether

The point (x,y) needs to be represented as(x,y,1) -> this is called Homogeneous

coordinates!

Matrix Representations and Homogeneous


38/93

Matrix Representations and HomogeneousCoordinates

Many applications involves sequences of geometric

transformations. For example, an animation might require an object to be

Translated, Rotated and Scaled at each increment of the

motion.

If you want to first rotates an object, then scales it, you can

combine those two transformation to a composite

transformation like the following equation:

0 cos sincos sin*

0 sin cossin cos

x x x

y y y

S S SA S R

S S S


39/93

However, it will be difficult to deal with the above composite

transformation and translation together. Because, translation is

not 2 by 2 matrix representation.

So ,here we consider how the matrix representations discussed in

the previous sections can be reformulated so that such

transformation sequences can be efficiently processed.


40/93

In fact, we can expressed three basic two-dimensional

transformation( translation, rotation, and scaling) in the

general matrix form:

Now this equation can be reformulated to eliminate the

matrix addition operation using Homogeneous Coordinates.

21

'MPMP

Homogeneous Coordinates


41/93

Homogeneous Coordinates

A standard technique for expanding each two-

dimensional coordinate position representation (x, y) tothree-element representation , calledHomogeneous Coordinates, where homogeneous h isa nonzero value such that

For geometric transformation, we can chose the

homogeneous parameter h to be any nonzero value.Thus there are an infinite number of equivalenthomogeneous representations for each coordinatepoint (x, y) .

( , , )h h

x y h

,h hx y

x yh h


42/93

A convenient choice is simply to set h=1, so each two-dimensional position is then represented with

homogeneous coordinate (x,y,1).

Expressing positions in homogenous coordinates allowsus to represent all geometric transformation equations as

matrix multiplications, which is the standard method usedin graphics systems. Two dimensional coordinate positionsare represented with three-elements column vectors, andtwo-dimensional transformation operations are expressed

as 3X3 matrices.

Homogeneous coordinate representation of 2D


43/93

Homogeneous coordinate representation of 2DTranslation

This translation operation can be written in theabbreviated form

11

010001

1

yx

tytx

yx

' ( , )x y

P T t t P

11

010001

1

yx

tytx

yx

Homogeneous coordinate representation of 2D


44/93

Homogeneous coordinate representation of 2DRotation

'( )P R P

This translation operation can be written in theabbreviated form

1100

0cossin

0sincos

1

y

x

y

x

Homogeneous coordinate representation of 2D Scaling


45/93

Homogeneous coordinate representation of 2D Scaling

1100

0000

1

yx

sysx

yx

'

( , )P S sx sy P

This scaling operation can be written in the abbreviatedform

Properties of homogeneous coordinate 2D


46/93

Properties of homogeneous coordinate 2Dtransformation.

(1) translation and rotation have additive properties.

(2) scaling transformation has multiplicative properties.

( , ) ( , ) ( , )x y h l x h y l

T t t T t t T t t t t

1 2 1 2( ) ( ) ( )R R R

( , ) ( , ) ( , )S sx st S sh sl S sx sh st sl


47/93

Composite Transformation

We can represent any sequence of

transformations as a single matrix. No special cases when transforming a point

matrix vector.

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?


48/93

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

hlgjhkgiflejfkei

dcba

lkji

hgfe

dcba

?


49/93

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

Composition Properties

?

dhcfdgce

bhafbgae

hg

fe

dc

ba

hdgbhcga

fdebfcea

dc

ba

hg

fe


50/93

Order of operations

So, it does matter. Lets look at an

example:1. Translate2. Rotate

1. Rotate2. Translate

C m it T f m ti M t i


51/93

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 origin3. Translate (pivot point is returned to original position)

T(pivot) R() T(pivot)1 0 -

tx0 1 -

ty0 0 1

cos -sin 0sin cos 00 0 1

1 0 tx0 1 ty0 0 1 . .

cos -sin -tx cos+ ty sin + txsin cos -tx sin - ty cos + ty

0 0 1

cos -sin -tx cos+ tysinsin cos -tx sin - tycos0 0 1

1 0 tx0 1 ty0 0 1 .


52/93

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

point (1,2). Find P?


cos -sin -tx cos+ ty sin + txsin cos -tx sin - ty cos + ty0 0 1

x

y1.

0.5 -0.866 -1.0.5 + 2.0.866 + 1

0.866 0.5 -1.0.866- 2.0.5 + 20 0 1

2

51.

0.5 - 0.866 2.2320.866 0.5 0.1340 0 1

251

. =-1.0984.366

1

P = (-1, 4)

Sin 60 = 0.8660

Kos 60 = 1/2

Without using composite homogenus


53/93

Without using composite homogenusmatrix

Example

Perform 90 rotation of a point P(5, 1) about a pivot point(2, 2). Find P?

1. Translate pivot point ke asalan ( tx = -2, ty = -2) Titik P(5, 1 ) P (3, -1)

2. Rotate P = 90 degree

P(3, -1) -- > kos 90 -sin 90 3 = 0 -1 3 =1

sin 90 kos 90 - 1 1 0 -13

3. Translate back ke pivot point (tx = 2 , ty = 2)

titik (1, 3 ) titik akhir (3, 5)


54/93

Composite Transformation MatrixGeneral Fixed-Point Scaling

Operation :-1. Translate (fixed point is moved to origin)2. Scale with respect to origin3. Translate (fixed point is returned to original position)

T(fixed) S(scale) T(fixed)

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

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


55/93

Answer1 0 -

tx0 1 -

ty0 0 1

Sx 0 00 Sy 0

0 0 1

1 0 tx0 1 ty

0 0 1 . .Sx 0 -tx Sx0 Sy -ty Sy0 0 1

1 0tx

0 1ty

0 0 1.

=

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

Sx 0 -tx Sx + tx0 Sy -ty Sy + ty0 0 1

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

.681

=10201

C it T f ti M t i


56/93


General Scaling Direction

Operation :-1. Rotate (scaling direction align with the coordinate axes)2. Scale with respect to origin3. Rotate (scaling direction is returned to original position)

R() S(scale) R()

Find the composite transformation matrixby yourself !!

Two-Dimensional Composite Transformation


57/93

Using matrix representations, we can set up asequence of transformation as a compositetransformation matrix, by calculating the product of theindividual transformations.

And, since many positions in a scene are typicallytransformed by the same sequence, its is moreefficient to first multiply the transformation matrices toform a single composite matrix.

So the coordinate position is transformed using thecomposite matrix M, rather than applying the individualtransformations M1 and then M2

'

2 1P M M P

M P

Composite Two-Dimensional Translations, Rotation


58/93

and Scaling:

( , ) ( , ) ( , )x y h l x h y lT t t T t t T t t t t

1 2 1 2( ) ( ) ( )R R R

( , ) ( , ) ( , )S sx st S sh sl S sx sh st sl

General Two-Dimensional Pivot-Point Rotation


59/93

So we can generate a 2D rotation about any other pivotpoint (x, y) by performing the following sequence oftranslate-rotate-translate operations.

(xr,yr) (xr,yr) (xr,yr)(xr,yr)

(1) Translate the object so that the pivot-point position


60/93

is moved to the coordinate origin.

(2) Rotate the object about the coordinate origin.

(3) Translate the object so that the pivot point isreturned to its original position.

The composite transformation matrix for thissequence is obtained with the concatenation:

1 0 1 0 cos sin (1 cos ) sincos sin 0

0 1 sin cos 0 0 1 sin cos (1 cos ) sin

0 0 10 0 1 0 0 1 0 0 1

r r r r

r r r r

x x x y

y y y x

Which can be expressed in the form


61/93

( , ) ( ) ( , ) ( , , )r r r r r r T x y R T x y R x y

Similarly, we can obtain the matrix representation of


62/93

General 2D Fixed-Point Scaling.( question ).

(1) Translate the object to so that the fixed point coincideswith the coordinate origin.

(2) Scale the object with respect to the coordinate origin.

(3) Use the inverse of the translation in step (1) to returnthe object to its original position.


63/93

100

)1(0

)1(0

100

10

01

100

00

00

100

10

01

yfy

xfx

f

fx

f

f

sys

sxs

y

x

s

s

y

x

y

Translate Scale Translate

(xr,yr) (xr,yr) (xr,yr) (xr,yr)

yxffffyxff ssyxSyxTssSyxT ,,,, ,,

We also obtain General 2D Scaling Directions.


64/93

Suppose we want to apply scaling factors withvalues specified by parameters s1 and s2 in the

directions shown in the following fig.

To accomplish the scaling without changing the orientationf h bj


65/93

of the object.(1) we perform a rotation so that the directions for s1 ands2 coincide with the x and y axes, respectively.

(2) Then the scaling transformation S(s1,s2) is applied.

(3) An opposite rotation to return points to their originalorientations.


66/93

100

0cossinsincos)(

0sincos)(sincos

)(),()(2

2

2

112

12

2

2

2

1

21

1

ssss

ssss

RssSR

For example, we turn a unit square into a parallelogramb t t hi it l th di l f (0 0) t (1 1)


67/93

by stretching it along the diagonal from (0,0) to (1,1).

x

y

(0,0)

(3/2,1/2)

(1/2,3/2) (2,2)

x

y

(0,0) (1,0)

(0,1) (1,1)

Scale

We first rotate the diagonal onto the y axis using angle45 d Th d bl it l th ith th li


68/93

45 degree. Then we double its length with the scalingvalues s1=1 and s2=2, and then we rotate again to returnthe diagonal to its original orientation.

Matrix Concatenation Properties


69/93

(1) Multiplication of matrices is associative.

(2) Transformation products, on the other hand, may not be

commutative.


70/93

Combined Tranformation

A number of transformations can be combined intoone 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

Combining Transformations


71/93

Combining Transformations(cont)

)(HHouseHdydxT

),(

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

1 2

3 4



72/93

Combining Transformations(cont)

The three transformation matrices are combinedas follows

1100

10

01

100

0cossin

0sincos

100

10

01

y

x

ty

tx

ty

tx

vdydxTRdydxTv ),()(),('

REMEMBER: Matrix multiplication is not

commutative so order matters



73/93

Co b g a s o at o s(cont)

The three transformation matrices are combinedas follows

vdydxTRdydxTv ),()(),('

1100

10

01

100

0cossin

0sincos

100

10

01

y

x

ty

tx

ty

tx

REMEMBER: Matrix multiplication is notcommutative so order matters


74/93

If Two Translations are applied then :

[ 1 0 tx1, 1 0 ty1, 0 0 1] * [ 1 0 tx2, 1 0 ty2, 0 0 1]then Result Is : [1 0 tx1+tx2, 1 0 ty1+ty2, 0 0 1].

If Two Rotations are applied then :-

[ cos (Q1+Q2) -sin(Q1+Q2) 0

sin (Q1+Q2) cos (Q1+Q2) 1

0 0 1 ]


75/93

If Two Scalings are Applied then :-

[ Sx1.Sx2 0 00 Sy1.y2 0

0 0 1]


76/93

Inverse Transformations

Transformations can easily be reversed usinginverse transformations

T-1= 1 0 -Tx

0 1 -Ty0 0 1

R-1 = Cos Q Sin Q 0

-Sin Q Cos Q 00 0 1


77/93

S-1 = 1/Sx 0 0

0 1/Sy 0

0 0 1


78/93

Inverse Transformation

Rotation About An Arbitrary Point :-[T] = [ Tr ] [Rq] [Tr -1 ]

where Tr = Translation Matrix

Rq = Rotation Matrix

Tr-1 = Inverse Translation Matrix

Scaling :Fixed Point

[T] [Sx][T-1]

Exercises 1


79/93

Exercises 1

x

y

0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(2, 3)

(3, 2)(1, 2)

(2, 1)

Translate the shape below by (7, 2)


80/93


81/93

Exercises 3

Rotate the shape below by 30 about the origin

x

y

0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(7, 3)

(8, 2)(6, 2)

(7, 1)


82/93

Exercise 4

Write out the homogeneous matrices for theprevious three transformations

______

______

______

______

______

______

______

______

______

Translation Scaling Rotation


83/93

Exercises 5

Using matrix multiplication calculate therotation of the shape below by 45 about its

centre (5, 3)

x

y

0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

(5, 4)

(6, 3)(4, 3)

(5, 2)

REFLECTION: Produce Mirror Image of An Object


84/93

REFLECTION: Produce Mirror Image of An Object.

Reflection with respect to the axis

X axis y axis original point

100

010

001

100

010

001

100

010

001

x

y 1

32

1

32 x

y1

32

1

3 2

x

y

3

1

3 2

1

2


85/93

Reflection X Axis

Image of an Object is formed on the sideopposite to where

the object is lying wrt mirror line.

Perpendicular distance is same from the mirror

line to the distance of reflected image.

X = x and Y = -y

[ 1 0

0 -1] [ x = [ 1 0 * [ x

y ] 0 -1] y]


86/93

Reflection Along Y Axis ( X = 0)

X = -x Y = y

[ -1 0

0 1 ]


87/93

Reflection About Origin

Mirror Line is Axis Perpendicular to xy plane andwill pass through origin .

When Reflection done , then both X and Y Coor.Flipped.

X = - X and Y = -Y.

Matrix :

[ x = [ -1 0 * [ x

y] 0 -1 ] y]

Reflection with respect to a line


88/93

100

001

010

x

y


89/93

Reflection About Line Y = - X

Line Y = -x becomes x = -y and y = -x. Represented as

[ x = [ 0 -1 * [ x

Y] -1 0] y]

Shear

Two common shearing transformations are those


90/93

Two common shearing transformations are thosethat shift coordinate x values and those that shift y

values. An x-direction shear relative to the x axis is produced

with the transformation matrix

100

010

01 xsh

x = x + shxy, y = y

x

y

x

y

(0,0) (1,0)

(1,1)(0,1)

(0,0) (1,0)

(3,1)(2,1)

Any real number can be assigned to the shear parametershx A coordinate position (x y) is then shifted horizontally


91/93

shx. A coordinate position (x, y) is then shifted horizontallyby an amount proportional to its perpendicular distance ( yvalue ) from the x axis.

We can generate x-direction shears relative to otherreference lines with yref=-1.

Now, coordinate positions are transformed as

x = x + shx(y-yref), y = y

(3,1)

1 refxx yshsh


92/93

The following Fig. illustrates the conversion of asquare into a parallelogram with shy=0.5 and yref=-1.

100

010

refxx

x

y

x

y

(0,0) (1,0)

(1,1)(0,1)

(1/2,0)(3/2,0)

(2,1)(1,1)

(0,-1)

Shearing operations can be expressed as a sequences ofbasic transformations. Such as a series of rotation and


93/93

scaling matrices

Documents

7. Transformation- 2D