Lecture11!11!16827 2D Transformations

7/23/2019 Lecture11!11!16827 2D Transformations

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Chapter 5

2-D Transformations


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October 13, 2015 Computer Graphics 2

Contents

1. Homogeneous coordinates

2. Matrices

3. Transformations

4. Geometric Transformations

5. Inverse Transformations

6. Coordinate Transformations

7. Composite transformations


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Homogeneous Coordinates

There are three t!pes of co"ordinate s!stems1. Cartesian Co-ordinate System

# Left Handed Cartesian Co-ordinate System$ C%oc&'ise(

# Right Handed Cartesian Co-ordinate System $ )nti C%oc&'ise(

2. Polar Co-ordinate System

3. Homogeneous Co-ordinate System

*e can a%'a!s change from one co"ordinate s!stem to

another.


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# ) point (x, ycan +e re"'ritten in homogeneouscoordinatesas (xh, yh, h

# The homogeneous parameterhis a non",ero va%ue suchthat-

# *e can then 'rite an! point (x, yas (hx, hy, h

# *e can convenient%! choose h ! 1so that(x, y+ecomes (x, y, 1

hxx h=

hyy h=


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Advantages:. Mathematicians use homogeneous coordinates as the!

a%%o' sca%ing factors to +e removed from e/uations.

2. )%% transformations can +e represented as 303 matrices

ma&ing homogeneit! in representation.

3. Homogeneous representation a%%o's us to use matri1mu%tip%ication to ca%cu%ate transformations e1treme%!efficient

4. ntire o+ect transformation reduces to sing%e matri1mu%tip%ication operation.

5. Com+ined transformation are easier to +ui%t andunderstand.


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Contents

. Homogeneous coordinates

2. Matrices

3. Transformations






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Matrices

Definition- ) matrixis an n m arra! of sca%ars arranged

conceptua%%! in n ro's and m co%umns 'here n and m are

positive integers. *e use ) and C to denote matrices.

If n 8 m 'e sa! the matri1 is a square matrix. *e often refer to a matri1 'ith the notation

A = a!i"#$%" 'here a$i( denotes the sca%ar in the ith ro' and

the th co%umn

9ote that the te1t uses the t!pica% mathematica% notation 'here

the i and are su+scripts. *e:%% use this a%ternative form as it is

easier to t!pe and it is more fami%iar to computer scientists.


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Matrices

&ca'ar-matrix mu'tip'ication:

) 8 ;a$i(s e%se'here Theinverse )". does not a%'a!s e1ist. If it does then

)".) 8 ) )".8 I

Given a matri1 ) and another matri1 'e can chec& 'hether

or not is the inverse of ) +! computing ) and ) and

seeing that ) 8 ) 8 I


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Matrices

# ach point ?$1!( in the homogenous matri1 form is

represented as

# @eca%% matri1 mu%tip%ication ta&es p%ace-

3333000

000

000

xxx"iyhxg

"fyexd

"#y$xa

"

y

x

ihg

fed

#$a

++

++

++

=

3

x

y

x


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Matrices

Matri1 mu%tip%ication does 9AT #ommute-

Matri1 composition 'or&s right-to-left.

# Compose-

# Then app%! it to a co%umn matri1 v-

It first appies Cto v, the! appies B to the resut, the! appies A to the resut of

that"

MN NM

v = Mv

v = ABC( )v

v = A

B

C

v( )( )

M = ABC


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Contents


2. Matrices

). Transformations






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Transformations

# ) transformationis a function that maps a point $or vector(into another point $or vector(.

# )n affine transformationis a transformation that maps %ines

to %ines.

# *h! are affine transformations BniceB

*e can define a po%!gon using on%! points and the %ine segments

oining the points.

To move the po%!gon if 'e use affine transformations 'e on%! must

map the points defining the po%!gon as the edges 'i%% +e mapped to

edges

# *e can mode% man! o+ects 'ith po%!gons"""and shou%d"""

for the a+ove reason in man! cases.


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Transformations

# )n! affine transformation can +e o+tained +! app%!ing inse/uence transformations of the form

Trans%ate

Dca%e

@otate @ef%ection

# Do to move an o+ect a%% 'e have to do is determine the

se/uence of transformations 'e 'ant using the 4 t!pes of

affine transformations a+ove.


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Transformations

# *eometric Transformations: In Geometric transformationan o+ect itse%f is moved re%ative to a stationar! coordinates!stem or +ac&ground. The mathematica% statement of thisvie' point is descri+ed +! geometric transformation app%ied

to each point of the o+ect.

# +oordinate Transformation: The o+ect is he%d stationar!

'hi%e coordinate s!stem is moved re%ative to the o+ect.These can easi%! +e descri+ed in terms of the oppositeoperation performed +! Geometric transformation.


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Transformations

# *hat does the transformation do

# *hat matri1 can +e used to transform the origina%

points to the ne' points

# @eca%%""" moving an o+ect is the same as changing a

frame so 'e &no' 'e need a 3 3 matri1

# It is important to remem+er the form of these

matrices


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Contents


2. Matrices

3. Transformations

,. *eometric Transformations





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Geometric Transformations

# In Geometric transformation an o+ect itse%f is moved re%ativeto a stationar! coordinate s!stem or +ac&ground. The

mathematica% statement of this vie' point is descri+ed +!

geometric transformation app%ied to each point of the o+ect.

Earious Geometric Transformations are-

Trans'ation

Dca%ing

@otation

@ef%ection Dhearing


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Geometric Transformations

#Trans'ation

#Dca%ing#@otation

#@ef%ection

#Dhearing


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Geometric Trans%ation

Is defined as the disp%acement of an! o+ect +! a given

distance and direction from its origina% position. In simp%e

'ords it moves an o+ect from one position to another.

x

! x % tx y

! y % ty

9ote- House shifts position re%ative to origin

y

x>

2

2

3 4 5 6 7 F >

3

4

5

6

E 8 t1I=t!J


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Geometric Trans%ation 1amp%e

y

x>

2

2

3 4 5 6 7 F >

3

4

5

6

$ ( $3 (

$2 3(

Trans'ation )(/20

$4 3(

$5 5(

$6 3(


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# To ma&e operations easier 2" points are 'ritten as

homogenous coordinate co%umn vectors

# The trans%ation of a point ?(x,y+! (tx, tycan +e 'ritten

in matri1 form as-

=

=

=

+==

...>>

.>

>.

($

y

x

Py

x

Pty

tx

&

'ty(tx)*hereP&P

)

)


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# @epresenting the point as a homogeneous co%umn vector

'e perform the ca%cu%ation as-

tyyytxxx

#om+aringon

tyy

txx

yx

tyyx

txyx

y

x

ty

tx

y

x

+=+=

+

+

=

++++

++

=

=

..0.0>0>

.00.0>

.00>0.

..>>

.>

>.

.


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Geometric Transformations#Trans%ation

#&ca'ing#@otation

#@ef%ection

#Dhearing


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Geometric Dca%ing

Dca%ing is the process of e1panding or compressing the

dimensions of an o+ect determined +! the sca%ing factor.

Dca%ar mu%tip%ies a%% coordinates

x

! Sx x y

! Sy y AT+H 3T:

# A+ects gro' and move

9ote- House shifts position re%ative to origin

y

x>

2

2

3 4 5 6 7 F >

3

4

5

6

2

3

3

6

3


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Geometric Dca%ing

# The sca%ing of a point ?(x,y+! sca%ing factors Sxand Sy

a+out origin can +e 'ritten in matri1 form as-

=

=

=

=

=

=

>>

>>

>>

>>

>>>>

($

ys

xs

y

x

s

s

y

x

thatsu#h

yx

Pyx

Pss

S

*herePSP

y

x

y

x

y

x

sysx

sysx


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Geometric Dca%ing 1amp%e

y

>

2

2

3 4 5 6 7 F >

3

4

5

6

$ ( $3 (

$2 3(

$2 2( $6 2(

$46(

&ca'e !2" 2$


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#Dca%ing

#4otation

#@ef%ection

#Dhearing


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Geometric @otation

# The rotation of a point ? $x,y( about origin +! specifiedang%e (/ counter c%oc&'ise( can +e o+tained as

x8x cos#y sin

y8x sin=y cos

# To rotate an o+ect 'e have to rotate a%% coordinates

6

=

y

x>

2

2

3 4 5 6 7 F >

3

4

5

6

x

$1!(

$1:!:(

5et us derive these equations


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Geometric @otation

+

=

=

=

=

=

=

cossin

sincos

>>

>cossin

>sincos

>>

>cossin>sincos

($

yx

yx

y

x

y

x

thatsu#h

yx

Pyx

PR

*herePRP

# The rotation of a point ?(x,y+! an ang%e a+out origincan +e 'ritten in matri1 form as-


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Geometric @otation 1amp%e

y

>

2

2

3 4 5 6 7 F >

3

4

5

6

$>>( $2>(

$2(


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#Dca%ing

#@otation

#4ef'ection

#Dhearing


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Geometric @ef%ection

# Mirror ref%ection is o+tained a+out "a1is

x8 x

y8 # y

# Mirror ref%ection is o+tained a+out "a1isx8 #x

y8yy

x>

2

2

3 4 5 6 7 F >

3

4

5

6

" "F "7 "6 "5 "4 "3 "2 "">


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# The ref%ection of a point ?(x,ya+out "a1is can +e 'ritten

in matri1 form as-

=

=

=

=

=

=

>>

>>

>>

>>

>>>>

($

y

x

y

x

y

x

thatsu#h

yx

Pyx

P0

*hereP0P

x

x


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# The ref%ection of a point ?(x,ya+out "a1is can +e 'ritten

in matri1 form as-

=

=

=

=

=

=

>>

>>

>>

>>

>>>>

($

y

x

y

x

y

x

thatsu#h

yx

Pyx

P0

*hereP0P

y

y


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# The ref%ection of a point ?(x,ya+out origin can +e 'ritten

in matri1 form as-

=

=

=

=

=

=

>>

>>

>>

>>

>>>>

($

y

x

y

x

y

x

thatsu#h

yx

Pyx

P0

*hereP0P

xy

xy


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#Dca%ing

#@otation

#@ef%ection

#&hearing


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Geometric Dhearing

# It us defined as ti%ting in a given direction

# Dhearing a+out !"a1is

x8 x % ay

y

8y % $xy

x>

2 3 4 5

2

3

$(

y

x>

2 3 4 5

2

3

a = 2

= )

4 $ 34(

$ 2(

$ 3(


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Geometric Dhearing

# The shearing of a point ?(x,yin genera% can +e 'ritten in

matri1 form as-

+

+

=

=

=

=

=

=

>>

>

>

>>

>>

($

6

6

$xy

ayx

y

x

$

a

y

x

thatsu#h

yx

Pyx

P$a

Sh

*herePShP

$a

$a


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Geometric Dhearing

# If + 8 > +ecomes Dhearing a+out "a1is

x8 x % ay

y8y

y

x>

2 3 4 5

2

3

$(

y

x>

2 3 4 5

2

3 a = 2

$ 2($ 3(


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Geometric Dhearing

# The shearing of a point ?(x,ya+out "a1is can +e 'ritten

in matri1 form as-

+

=

=

=

=

=

=

>>

>>

>

>>

>>>

($

>6

>6

y

ayx

y

xa

y

x

thatsu#h

yx

Pyx

Pa

Sh

*herePShP

a

a


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Geometric Dhearing

# If a 8 > it +ecomes Dhearing a+out !"a1is

x8 x

y8y % $x

y

x>

2 3 4 5

2

3

$(

y

x>

2 3 4 5

2

3 = )

4

$ 3(

$ 4(


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Geometric Dhearing

# The shearing of a point ?(x,ya+out "a1is can +e 'ritten

in matri1 form as-

+=

=

=

=

=

=

>>

>

>>

>>

>>>

($

6>

6>

$xy

x

y

x

$y

x

thatsu#h

yx

Pyx

P$Sh

*herePShP

$

$


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Contents


2. Matrices mu%tip%ications

3. Transformations


6. (nverse Transformations




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Inverse Transformations

# (nverse Trans'ation- isp%acement in direction of #E

# (nverse &ca'ing- ivision +! Sxand Sy

==

>>

>

>

ty

tx

&& ))

==

.>>

>.>

>>.

J.6J.

.

6 y

x

sysxsysx S

S

SS


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Inverse Transformations

# (nverse 4otation- @otation +! an ang%e of #

# (nverse 4ef'ection- @ef%ect once again

==

>>

>>

>>

xx 00

==

>>

>cossin

>sincos

RR


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Contents



3. Transformations



7. +oordinate Transformations



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Coordinate Transformations

#+oordinate Transformation: The o+ect is he%dstationar! 'hi%e coordinate s!stem is moved re%ative

to the o+ect. These can easi%! +e descri+ed in terms

of the opposite operation performed +! Geometric

transformation.


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# +oordinate Trans'ation- isp%acement of the coordinate

s!stem origin in direction of #E

# +oordinate &ca'ing- Dca%ing an o+ect +! Sxand Sy or

reducing the sca%e of coordinate s!stem.

== .>>

.>

>.

ty

tx

&& ))

==

>>

>>>>

J6J6 y

x

sysxsysx SS

SS


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# +oordinate 4otation- @otating Coordinate s!stem +! an

ang%e of #

# +oordinate 4ef'ection- Dame as Geometric @ef%ection

$'h!(

==

>>

>>

>>

xx 00

==

>>

>cossin

>sincos

RR


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Contents



3. Transformations




8. +omposite transformations


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Composite Transformations

# ) num+er of transformations can +e com+ined into one matri1 to

ma&e things eas!

)%%o'ed +! the fact that 'e use homogenous coordinates

# Matri1 composition 'or&s right-to-left.

Compose-

Then app%! it to a point-

It first app%ies +to v" then app%ies 9 to the resu%t then app%ies A to the resu%t of that.

M = A

B

C

v = Mv

v = ABC( )vv = A B Cv( )( )


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4otation aout Aritrar oint !h";$

# Imagine rotating an o+ect around a point $h&( other than

the origin

Trans%ate point $h&( to origin

@otate around origin

Trans%ate +ac& to point


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Composite TransformationsKet ? is the o+ect point 'hose rotation +! an ang%e a+out the fi1ed point $h&(is to +e found. Then the composite transformation @6$h6&(can +e o+tained +!

performing fo%%o'ing se/uence of transformations -

. Trans%ate $h&( to origin and the ne' o+ect point is found as

?8 TE$?( 'here E8 # hI # &L

2. @otate o+ect a+out origin +! ang%e and the ne' o+ect point is

?28 @$?(

3. @etrans%ate $h&( +ac& the fina% o+ect point is

?M8 T"E$?2( 8 T"E $?2(

The composite transformation can be obtained by back substituting

?M

8 T"E$?2

(8 T"E@$?(

8 T"E@TE$?( 'here E 8 # hI # &L

Thus 'e form the matri1 to +e @$h&(8 T"E@TE


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# The composite rotation transformation matri1 is

+

++

=

=

>>

cossincossin

sincossincos

>>

>

>

>>

>cossin

>sincos

>>

>

>

(6$6

11h

h1h

1

h

1

h

R 1h

REMEMBER:#atri$ mutipicatio! is !ot

commutati%e so or&er matters


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&ca'ing aout Aritrar oint !h";$

# Imagine sca%ing an o+ect around a point $h&( other than

the origin

Trans%ate point $h&( to origin

Dca%e around origin

Trans%ate +ac& to point


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# The composite sca%ing transformation matri1 is

+

+

=

=

>>

.>

.>

>>

>

>

>>

>>

>>

>>

>

>

(6$66

1sy1sy

hsxhsx

1

h

sy

sx

1

h

S 1hsysx


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1ercises

x

y

>

2

2

3 4 5 6 7 F >

3

4

5

6

'2, 3(

'3, 2('1, 2(

'2, 1(

Trans%ate the shape +e%o' +! $7 2(


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1ercises 2

x

y

>

2

2

3 4 5 6 7 F >

3

4

5

6

'2, 3(

'3, 2('1, 2(

'2, 1(

Dca%e the shape +e%o' +! 3 inxand 2 iny


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1ercises 3@otate the shape +e%o' +! 3>N a+out the origin

x

y

>

2

2

3 4 5 6 7 F >

3

4

5

6

'2, 3(

'3, 2('1, 2(

'2, 1(


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1ercise 4

*rite out the homogeneous matrices for the previous

three transformations

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

)ra!satio! *cai!+ otatio!


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1ercises 5

Psing matri1 mu%tip%ication ca%cu%ate the rotation of theshape +e%o' +! 45N a+out its centre $5 3(

x

y

>

2

2

3 4 5 6 7 F >

3

4

5

'5, 4(

'6, 3('4, 3(

'5, 2(


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1ercise 6

@otate a triang%e )C )$>>( $( C$52( +! 45>

. )+out origin $>>(

2. )+out ?$""(

=

>>

>2222

>2222>

45R

=>>

22222

2222

($45>

R

=2>

5>


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1ercise 7

Magnif! a triang%e )C )$>>( $( C$52( t'ice &eeping

point C$52( as fi1ed.

=>>

22>

5>2

(25$22S

=2>2

535

5>


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1ercise F

Ket ? +e the o+ect point

'hose ref%ection is to ta&en

a+out %ine K that ma&es anang%e 'ith =ve "a1isand has intercept as

$>+(. The composite

transformation MK can +efound +! app%!ing

fo%%o'ing transformations

in se/uence-X-axis

-.m$/b

Y-axis

'0,b(

'$,(

'$,(

Descrie transformation M5


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1ercise F. Trans%ate $>+( to origin so that %ine passes through origin and

? is transformed as

?I8 TE$?( 'here E 8 #hI #&L

2. @otate +! an ang%e of #so that %ine a%igns 'ith =ve "a1is

?II8 @"$?I(

3. 9o' ta&e mirror ref%ection a+out "a1is.

?III8 M1$?II(

4. @e"rotate %ine +ac& +! ang%e of

?IE8 @$?III(5. @etrans%ate $>+( +ac&.

?8 T"E$?IE(


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1ercise FThe composite transformation can be obtained by back

substituting

?8 T"E$?IE(

8 T"E.@$?III(

8 T"E.@. M1$?II(8 T"E.@. M1 .@"$?I(

8 T"E.@. M1 .@". TE$?(

Thus 'e form the matri1 to +e MK8 T"E.@. M1 .@". TE

'here E 8 #>.I #+.L


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1ercise F

+

=

=

=

=

>>

(sin$coscossincossin2

cossin2cossin2sincos>>

coscossin

sinsincos

>>

cossin

>sincos

>>

coscossin

sinsincos

>>

>>

>>

>>

cossin

>sincos

>>

>

>>

>>

>cossin

>sincos

>>

>>

>>

>>

>cossin

>sincos

>>

>

>>

2222

22

$$

$

$

$

$

$

$

$

$$0L


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1ercise F

...

>>

2

2

2

2

C($tan6tan

tan2sin6

tan

tan2cos

>>

2cos2cos2sin

2sin2sin2cos

>>

(sin$coscossincossin2

cossin2cossin2sincos

22

2

2

222

2

2

2

2

2

2222

22

0&Cm

$

m

m

m

mm

$m

m

m

m

m

*hymand+utting

$$

$

$$

$

0L

++

+

+

++

=

=+

=+

=

+

=

+

=


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1ercise

@ef%ect the diamond shaped po%!gon 'hose vertices are )$">(

$>"2( C$>( and $>2( a+out

. Hori,onta% Kine !82

2.Eertica% Kine 1 8 2

3. Kine K- !81=2.

==

>>

4>

>>

2y0

==

>>

>>

4>

2x0

=+=

>>2>

2>

2xy

0


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1ercise >

A+tain ref%ection a+out Kine ! 8 1

=

=

=

>>

>>

>>

>>

>>

>>

>>

>cossin

>sincos

@erotateanda1isa+outref%ectionta&e45"+!@otate-

@erotateanda1isa+outref%ectionta&e45+!@otate-

>>

>>

4545

>

4545

>

yx

x

y

00RHere

R0R

0ethod

R0R

0ethod

==

>>

>>

>>

xy0


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1ercise

?rove that

a. T'o successive trans%ations are additive Jcommutative.

+. T'o successive rotations are additive Jcommutative.

c. T'o successive Dca%ing are mu%tip%icative Jcommutative.

d. T'o successive ref%ections are nu%%ified JInverti+%e.

Is Trans%ation fo%%o'ed +! @otation e/ua% to @otation fo%%o'ed

+! trans%ation


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1ercise a. T


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1ercise A'so"

e#ommutati)arenstranslatiosu##essi)et*oHen#e

&&&

tyty

txtx

tyty

txtx

tytx

tytx

.&&&

))))

))))

.

>>

:>

:>

>>

:>

:>

>>

:>:>

.

>>

>>

::

::

==

+

+

=

+

+

=

==

+

+


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1ercise . T


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1ercise A'so"

e#ommutati)ares#alingssu##essi)et*oHen#e

.SSS

sysy

sxsx

sysy

sxsx

sysx

sysx

.SSS

sysxsysxsysysxsx

sysxsysxsysysxsx

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79/84


1ercise c. Tcommutative.

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2

>>

>(cos$(sin$

>(sin$(cos$

>>

>cossin

>sincos

.

>>

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.+!rotation+!fo%%o'ed+!rotationtheformu%atefirst*e

andang%e+!descri+edarerotationsKet t'o

22

22

22

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2

2

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80/84


1ercise A'so"

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22

2

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>(cos$(sin$

>(sin$(cos$

>>

>(cos$(sin$

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>>

>cossin>sincos

.

>>

>cossin>sincos

22

22

22

22

22

22

==

++

++

=

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=

+


81/84


1ercise d. T>

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82/84

* t h


83/84


*cratch

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y

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2

2

3 4 5 6 7 F >

3

4

5

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84/84

Lecture11!11!16827 2D Transformations

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