HÌNH HỌC CAO CẤP GV: TRẦN LÊ NAM
CHƯƠNG 2: CÁC PHÉP BIẾN HÌNH TRONG
MẶT PHẲNG
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85C:8?56"#$"5F1&(/C5F1&( J"#$"%&:V
(4i&(2#C=

I2TU"#$":R#S(_
P P
→
G"#$"%&:V(4i&(2#5F
1&(56"#$"%&:V(4i&(2#:R#S(
e
r

P P→
ur ur
=#] 7i.KLCMN.
KLCMN.
MN
uuur
L
u
r
 6
' 'M N
uuuuur
L
u
r
.F
( )
f u
ur r
L
u
r

.T*:G
f
ur
56"#$":R#S(
e
r

P P→
ur ur
1.6 Xác định ánh xạ afin
f
HÌNH HỌC CAO CẤP GV: TRẦN LÊ NAM
Định lí :
#*.K
∈
2
A
.
ϕ

2 2
V V→
M
2
V
L
A
ur
N.1#:G(R(T4i#S(+D(9#Z/C
C()F

2
A
E/*#*
 CMNLK

( )
MN
ϕ
uuur
L
( ) ( )
f M f N
uur ur
.
∀
..
∈
2
A
/?
ϕ
569#Z(4i&(2#5F1&( J9#ZC
#[+#Z$(9#ZC
P P
→
%&+>:;+(#6#KE/*#*
( )
f IM
ur uuur
L

' 'I M
uuuuur
=/#[+#)nC56"#$"%&/C=
#7( 7i. J%/:;+%S(1'..(#-#6 6M..NL1=
/G
1
IA k IB
IC
k
−
=
−
uur uur
uur
d#:G.?KLCMN.KLCMN.KLCMN.G

( )
f IA
ur uur
L
' 'I A
uuuur
`
( )
f IB
ur uur
L
' 'I B
uuuur
`

( )
f IC
ur uur
L
' 'I C
uuuur
=
   r* 
f
ur
56"#$"%&:V(4i&(2#F 
' ' ' '
' '
1
I A kI B
I C
k
−
=
−
uuuur uuuur
uuuur
=!4:G
#[(x/)nK.K.K(#-#6 6MK.K.KNL1=
  K K
K
 K 
7iC56"#$"/C 6)y)6CMNLK
m
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IJ#/:;+%S(1'.</ 6K.K56O#</#WP4/C(/
G
( )
f MN
ur uuur
L
f
ur
M
IN
uur
a
IM
uuur
NL
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uuuur
a
' 'I M
uuuuur
L
' 'M N
uuuuur
=I7iC56"#$"/C</
5F1&( J"#$"%&:V(4i&(2#TX#*
#$"/C#()F56T4i#S(=([56.OElG"#$"/CCK</
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f
ur
:X#*.(#'CK()z

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:G. 'C 6CKz5F1&( J"#$"%&:V(4i&(2#
f
ur
F
( )
f I M
ur uur
L
' 'I M
uuuuur
`
( )
f IM
ur uuur
L
' "I M
uuuuur
.E4i)/KKLK. 7iCKLC=
Định lí :
C
2 2
A A→

'M Ma

' 'I M
uuuuur
L
( )

IM A
ϕ
∈
uuur ur
#[+#Tp(#Sif(I) = I’

( ) ( )f M f N
uuuuuuuuuuur
L
' ( )I f N
uuuuur
a
' ( )I f M
uuuuur
L
( ) ( )f I f N
uuuuuuuuur
a
( ) ( )f I f M
uuuuuuuuuur
L
( )IN
ϕ
uur
a
( )IM
ϕ
uuur
L
(IN

ϕ
uur
a
)IM
uuur
L
( )MN
ϕ
uuur
.
2
, ,M N A∀ ∈
MdNLd
⇔
PM
uuur
L
K PN
uuur
h
HÌNH HỌC CAO CẤP GV: TRẦN LÊ NAM

( )PM
ϕ
⇔
uuur
L
( )K PN
ϕ
uuur


' 'P M⇔
uuuuur
L
' 'k P N
uuuuur

⇒
MM’N’P’) = K
OEl
∃
f’
' 'A A→
(#x/+X:A#52
∀

∈
2
A

( ) '( )f M f M
uuuuuuuuuuuur
L
( ) 'f M I
uuuuur
v
' '( )I f M
uuuuuur
L
( ) ( )f M f I

uuuuuuuuuur
v
'( ) '( )f I f M
uuuuuuuuuuur
L
( )MI
ϕ
uuur
v
( )IM
ϕ
uuur
L
(MI
ϕ
uur
v
)IM
uuur
L
0
r
Hệ quả:
D(9#Z/C()F+,("#-
2
A
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:8%/:;+1#@(#-#6.#{/&4.. 6K.K.K56#/
(/+9()F+,("#-
2

A
(#'(R(T4i#S("#$"%&:V/C%&
(/+9(#6#(/+9KKK=
1.7 Biểu thức tọa độ của phép biến hình afin
      #*"#$"/CC 
P P
→
.5F1&(  J"#$"%&:V(4i&(2#
:f P P→
ur ur ur
/#?+U(F4/C
( )
; ,O i j
r r
6*:G</+,("#-=?KL
CMN. 69 $(
' ( ), ' ( )i f i j f j= =
r ur r ur r
.G(?/:D56
KLM"`PN.
' ( ; ), ' ( ; )i a b j c d= =
r ur
=
[56
OO ' ; ' ;pi q j i ai b j j ci d j= + = + = +
uuur r r r r r r r r
|
HÌNH HỌC CAO CẤP GV: TRẦN LÊ NAM
\i^(/OEl:Y J+U(F4:X#?.+D(:;+%S(1'G
(?/:D 6O#</GKLCMNG(?/:DMZK.iKN=/#Xi('+E]5F#3

{/MZ.iN 6MZK.iKN
/G
y j, ' y j'OM xi OM xi= + = +
uuur r r uuur r r
q4i)/
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uuuuur ur uuur ur r r r r r r
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uuuuur r r

( )
∗
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' ax+cy+p
'
x

y bx dy q
=


= + +

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r
6
j
r
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b d
 
 ÷
 
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b d
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b d
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 ÷
 
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;4(#[(?/:D</"#$"/CCG(#; &(TJT+/()7

'
'
x
y
 
 ÷
 
L
a c
b d
 
 ÷
 
'
'
x
y
 
 ÷
 
v
p

q
 
 ÷
 
}
HÌNH HỌC CAO CẤP GV: TRẦN LÊ NAM
/i
'
'
x
y
 
 ÷
 
L
'
'
x
y
 
 ÷
 
v
p
q
 
 ÷
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