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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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October 13, 2015 Computer Graphics 3
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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October 13, 2015 Computer Graphics 4
Homogeneous Coordinates
# ) 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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October 13, 2015 Computer Graphics 5
Homogeneous Coordinates
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
4. Geometric Transformations
5. Inverse Transformations
6. Coordinate Transformations
7. Composite transformations
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October 13, 2015 Computer Graphics 7
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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October 13, 2015 Computer Graphics 8
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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October 13, 2015 Computer Graphics 12
Contents
. Homogeneous coordinates
2. Matrices
). Transformations
4. Geometric Transformations
5. Inverse Transformations
6. Coordinate Transformations
7. Composite 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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October 13, 2015 Computer Graphics 17
Contents
. Homogeneous coordinates
2. Matrices
3. Transformations
,. *eometric Transformations
5. Inverse Transformations
6. Coordinate Transformations
7. Composite 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(
$6 3(
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Geometric Trans%ation
# To ma&e operations easier 2" points are 'ritten as
homogenous coordinate co%umn vectors
# The trans%ation of a point ?(x,y+! (tx, tycan +e 'ritten
in matri1 form as-
=
=
=
+==
...>>
.>
>.
($
y
x
Py
x
Pty
tx
&
'ty(tx)*hereP&P
)
)
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Geometric Trans%ation
# @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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Geometric Transformations#Trans%ation
#Dca%ing
#4otation
#@ef%ection
#Dhearing
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October 13, 2015 Computer Graphics 29
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>(
$2(
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Geometric Transformations#Trans%ation
#Dca%ing
#@otation
#4ef'ection
#Dhearing
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Geometric @ef%ection
# Mirror ref%ection is o+tained a+out "a1is
x8 x
y8 # y
# Mirror ref%ection is o+tained a+out "a1isx8 #x
y8yy
x>
2
2
3 4 5 6 7 F >
3
4
5
6
" "F "7 "6 "5 "4 "3 "2 "">
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October 13, 2015 Computer Graphics 34
Geometric @ef%ection
# The ref%ection of a point ?(x,ya+out "a1is can +e 'ritten
in matri1 form as-
=
=
=
=
=
=
>>
>>
>>
>>
>>>>
($
y
x
y
x
y
x
thatsu#h
yx
Pyx
P0
*hereP0P
x
x
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Geometric @ef%ection
# 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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Geometric @ef%ection
# 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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Geometric Transformations#Trans%ation
#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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October 13, 2015 Computer Graphics 41
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
. Homogeneous coordinates
2. Matrices mu%tip%ications
3. Transformations
4. Geometric Transformations
6. (nverse Transformations
6. Coordinate Transformations
7. Composite transformations
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October 13, 2015 Computer Graphics 45
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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October 13, 2015 Computer Graphics 46
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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October 13, 2015 Computer Graphics 47
Contents
. Homogeneous coordinates
2. Matrices mu%tip%ications
3. Transformations
4. Geometric Transformations
5. Inverse Transformations
7. +oordinate Transformations
7. Composite transformations
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October 13, 2015 Computer Graphics 48
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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Coordinate Transformations
# +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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Coordinate Transformations
# +oordinate 4otation- @otating Coordinate s!stem +! an
ang%e of #
# +oordinate 4ef'ection- Dame as Geometric @ef%ection
$'h!(
==
>>
>>
>>
xx 00
==
>>
>cossin
>sincos
RR
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October 13, 2015 Computer Graphics 51
Contents
. Homogeneous coordinates
2. Matrices mu%tip%ications
3. Transformations
4. Geometric Transformations
5. Inverse Transformations
6. Coordinate Transformations
8. +omposite transformations
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Composite Transformations
# ) num+er of transformations can +e com+ined into one matri1 to
ma&e things eas!
)%%o'ed +! the fact that 'e use homogenous coordinates
# Matri1 composition 'or&s right-to-left.
Compose-
Then app%! it to a point-
It first app%ies +to v" then app%ies 9 to the resu%t then app%ies A to the resu%t of that.
M = A
B
C
v = Mv
v = ABC( )vv = A B Cv( )( )
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Composite Transformations
4otation aout Aritrar oint !h";$
# 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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October 13, 2015 Computer Graphics 55
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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Composite Transformations
# 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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Composite Transformations
&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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Composite Transformations
# The composite sca%ing transformation matri1 is
+
+
=
=
>>
.>
.>
>>
>
>
>>
>>
>>
>>
>
>
(6$66
1sy1sy
hsxhsx
1
h
sy
sx
1
h
S 1hsysx
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October 13, 2015 Computer Graphics 60
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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October 13, 2015 Computer Graphics 64
1ercises 5
Psing matri1 mu%tip%ication ca%cu%ate the rotation of theshape +e%o' +! 45N a+out its centre $5 3(
x
y
>
2
2
3 4 5 6 7 F >
3
4
5
'5, 4(
'6, 3('4, 3(
'5, 2(
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October 13, 2015 Computer Graphics 65
1ercise 6
@otate a triang%e )C )$>>( $( C$52( +! 45>
. )+out origin $>>(
2. )+out ?$""(
=
>>
>2222
>2222>
45R
=>>
22222
2222
($45>
R
=2>
5>
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October 13, 2015 Computer Graphics 66
1ercise 7
Magnif! a triang%e )C )$>>( $( C$52( t'ice &eeping
point C$52( as fi1ed.
=>>
22>
5>2
(25$22S
=2>2
535
5>
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October 13, 2015 Computer Graphics 67
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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October 13, 2015 Computer Graphics 68
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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October 13, 2015 Computer Graphics 69
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$?(
Thus 'e form the matri1 to +e MK8 T"E.@. M1 .@". TE
'here E 8 #>.I #+.L
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October 13, 2015 Computer Graphics 70
1ercise F
+
=
=
=
=
>>
(sin$coscossincossin2
cossin2cossin2sincos>>
coscossin
sinsincos
>>
cossin
>sincos
>>
coscossin
sinsincos
>>
>>
>>
>>
cossin
>sincos
>>
>
>>
>>
>cossin
>sincos
>>
>>
>>
>>
>cossin
>sincos
>>
>
>>
2222
22
$$
$
$
$
$
$
$
$
$$0L
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October 13, 2015 Computer Graphics 71
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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October 13, 2015 Computer Graphics 72
1ercise
@ef%ect the diamond shaped po%!gon 'hose vertices are )$">(
$>"2( C$>( and $>2( a+out
. Hori,onta% Kine !82
2.Eertica% Kine 1 8 2
3. Kine K- !81=2.
==
>>
4>
>>
2y0
==
>>
>>
4>
2x0
=+=
>>2>
2>
2xy
0
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October 13, 2015 Computer Graphics 73
1ercise >
A+tain ref%ection a+out Kine ! 8 1
=
=
=
>>
>>
>>
>>
>>
>>
>>
>cossin
>sincos
@erotateanda1isa+outref%ectionta&e45"+!@otate-
@erotateanda1isa+outref%ectionta&e45+!@otate-
>>
>>
4545
>
4545
>
yx
x
y
00RHere
R0R
0ethod
R0R
0ethod
==
>>
>>
>>
xy0
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October 13, 2015 Computer Graphics 74
1ercise
?rove that
a. T'o successive trans%ations are additive Jcommutative.
+. T'o successive rotations are additive Jcommutative.
c. T'o successive Dca%ing are mu%tip%icative Jcommutative.
d. T'o successive ref%ections are nu%%ified JInverti+%e.
Is Trans%ation fo%%o'ed +! @otation e/ua% to @otation fo%%o'ed
+! trans%ation
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October 13, 2015 Computer Graphics 75
1ercise a. T
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October 13, 2015 Computer Graphics 76
1ercise A'so"
e#ommutati)arenstranslatiosu##essi)et*oHen#e
&&&
tyty
txtx
tyty
txtx
tytx
tytx
.&&&
))))
))))
.
>>
:>
:>
>>
:>
:>
>>
:>:>
.
>>
>>
::
::
==
+
+
=
+
+
=
==
+
+
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October 13, 2015 Computer Graphics 77
1ercise . T
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October 13, 2015 Computer Graphics 78
1ercise A'so"
e#ommutati)ares#alingssu##essi)et*oHen#e
.SSS
sysy
sxsx
sysy
sxsx
sysx
sysx
.SSS
sysxsysxsysysxsx
sysxsysxsysysxsx
:::.:.
:::.:.
>>
>:.>
>>:.
>>
>:.>
>>:.
>>
>:>>>:
.
>>
>>>>
==
=
=
==
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October 13, 2015 Computer Graphics 79
1ercise c. Tcommutative.
addit)earerotationssu##essi)et*oHen#e
R
.RR
2
2
>>
>(cos$(sin$
>(sin$(cos$
>>
>cossin
>sincos
.
>>
>cossin
>sincos
.+!rotation+!fo%%o'ed+!rotationtheformu%atefirst*e
andang%e+!descri+edarerotationsKet t'o
22
22
22
22
2
2
+=
++
++
=
=
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October 13, 2015 Computer Graphics 80
1ercise A'so"
e#ommutati)arerotationssu##essi)et*oHen#e
.RRR
.RR
22
2
>>
>(cos$(sin$
>(sin$(cos$
>>
>(cos$(sin$
>(sin$(cos$
>>
>cossin>sincos
.
>>
>cossin>sincos
22
22
22
22
22
22
==
++
++
=
++
++
=
=
+
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October 13, 2015 Computer Graphics 81
1ercise d. T>
>>
>>
>>
>>
>>
.
>>
>>
>>
a1is"a+outref%ectionconsiderusKet
=
=
=
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* t h
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October 13, 2015 Computer Graphics 83
*cratch
x
y
>
2
2
3 4 5 6 7 F >
3
4
5
6
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