< K1 ç ¡ AFFINE M È Ñ

(1)

< K1 ç ¡ AFFINE M È Ñ

^

ïrB

z

<

l

<Æ>h:r, Ñþ6 xC $, §<Æ½¨, 2003

(2)

+ ä

q

affine /BNçß A³_ lïr 7'/BNçß V³_ 1 ¨8LAü< ¨î's1lx Tb_

½

+Ë$í T_bL_A\¦ A³_affine ¨8s ôÇ. (> LA_ '§>= = A)

(3)

+ ä

P 1.7

A³_ affine ¨8T : A³→ A³(T : [x₁, x₂, x₃] 7→ [x⁰₁, x⁰₂, x⁰₃])H6£§ d

Ü¼Ð ³ðr)a.





 x⁰₁ x⁰₂ x⁰₃







=







a₁₁ a₁₂ a₁₃ a21 a22 a23

a31 a32 a33











 x₁ x2

x3





 +





 b₁ b2

b3







é

ß, '§>= A = (a_ij)Hq:£¤s¦ b = ^t(b₁, b₂, b₃).

'

§>= A = (a_ij)Hq:£¤siff det A 6= 0.

(4)

û B' å

{e1, e2, e3} V³_ ôÇ l e__ 7' xH x = x1e1+ x2e2+ x3e3

Ð ³ðr)a.

V³_ 1 ¨8LA\ @/ # e1, e2, e3 6£§õ °ú s ¨8)a¦ :











LA(e1) = a11e1+ a21e2+ a31e3

LA(e2) = a12e1+ a22e2+ a32e3

L_A(e₃) = a₁₃e₁+ a₂₃e₂+ a₃₃e₃

#

l"f L^AH 1 ¨8sÙ¼Ð '§>= A = (aij)Hq:£¤ss¦

L (x)= L (x e + x e + x e ) =x L (e ) + x L (e ) + x L (e )

(5)

s

M:, 7' x 1 ¨8L_A\ _ #

L_A(x) = y = y₁e₁+ y₂e₂+ y₃e₃

Ð ¨8)a¦











y1= a11x1+ a21x2+ a31x3

y2= a12x1+ a22x2+ a32x3

y3= a13x1+ a23x2+ a33x3

6£§, 7' b = ^t(b1, b2, b3)ëßpu_ ¨î's1lx Tb\ _ # A³_ &h [y₁, y₂, y₃]\¦&h [x⁰₁, x⁰₂, x⁰₃]\ s1lx

Tb[y1, y2, y3] = [x⁰₁, x⁰₂, x⁰₃].

∴ x⁰1= y1+ b1, x⁰₂= y2+ b2, x⁰₃= y3+ b3. Õ

ªQÙ¼Ð x⁰ = T(x) = T_bL_A(x) = Ax + bs.

(6)

¹M 1

A²_ affine ¨8

T :







x⁰₁= x1+ x2+ 1 x⁰₂= x2+ 2

\

_ # 6£§Ér#QbG> H?

(1) A = [0, 0], B = [1, 0], C = [1, 1], D = [0, 1]

(2) "é¶: x²₁+ x²₂= 1

(7)

b ô >T

(1) A⁰= B⁰=

C⁰ = D⁰=

(8)

b ô >T

(2)







x1= x⁰₁− x⁰₂+ 1 x2= x⁰₂− 2

`

¦ x²₁+ x²₂= 1\ @/{9.

(9)

¹M 2

A²(s Ðl\"fH E²)_ affine ¨8\ _ # ¿º &h s_ oH

ôÇ.

û B' å

f

§ýa³ð>\"f ¿º &h P = [x1, x2], Q = [y1, y2]s_ oH d(P, Q) =p(y₁− x₁)²+ (y₂− x₂)²

s

. A²_ affine ¨8`¦







x⁰₁= ax1+ bx2

x⁰₂= cx1+ dx2

(ad − bc 6= 0)s







y₁⁰ = ay₁+ by₂

y₂⁰ = cy₁+ dy₂ s. (P⁰= [x⁰₁, x⁰₂], Q⁰= [y₁⁰, y⁰₂])

(10)

(y₁⁰ − x⁰₁)²= {a(y₁− x₁) + b(y₂− x₂)}² (y₂⁰ − x⁰₂)²= {c(y1− x1) + d(y2− x2)}²

∴ (y₁⁰ − x⁰₁)²+ (y₂⁰ − x⁰₂)²= (a²+ c²)(y1− x1)²+ (b²+ d²)(y2− x2)² + 2(ab + cd)(y1− x1)(y2− x2)

7

£¤, {9ìøÍ&hÜ¼Ðd(P, Q) = d(P⁰, Q⁰)s $íwnôÇ¦ ½+É Ãº \O.

(11)

+ ä

P 1.8

affine ¨î A²_ 4P QR, 4P⁰Q⁰R⁰\"f

P → P⁰, Q → Q⁰, R → R⁰

affine ¨8Éréß >rFôÇ.

û B' å

7'−−→ P Q,−→

P RÉr 1 1lqwn.

7'−−−→ P⁰Q⁰,−−→

P⁰R⁰Ér 1 1lqwn.

7'/BNçß V²\"f l {−−→ P Q,−→

P R}\¦l {−−−→ P⁰Q⁰,−−→

P⁰R⁰}Ð `lH 1 ¨8 L_A(det A 6= 0)HìøÍ×¼r >rFôÇ. Õªo¦ &h P \¦ P⁰Ü¼Ð Ð?/H

¨î

's1lxTb(b =−−→

P P⁰) éß >rFôÇ. ÕªQÙ¼Ð ½¨ H

¨8ÉrTbLA(affine ¨8)s.

(12)

+ ä

P 1.9

affine /BNçß A³_ ^ P QRS, P⁰Q⁰R⁰S⁰\"f

P → P⁰, Q → Q⁰, R → R⁰, S → S⁰

affine ¨8Éréß >rFôÇ.

û B' å

&

ño 1.8õ q5pw.

(13)

¹M 3

A²\"f

[1, 0] → [2, −1], [0, 1] → [3, −4], [1, 1] → [4, −2]

affine ¨8`¦½¨ r¸.

b ô >T

½

¨ ¦ H affine ¨8`¦







x⁰₁= ax₁+ bx₂+ m

x⁰₂= cx₁+ dx₂+ n s ,

(14)

+ ä

P 1.10

e

__ 1 ¨8LAü< ¨î's1lx Tb\ @/ # L_AT_b= T_cL_A

T_c >rFôÇ. (ÅÒ_ : L^ATb6= TbLA)

û B' å

e

__ 7' x\ @/ #

(LATbL⁻¹_A )(x) = LATb(A⁻¹x) = LA(A⁻¹x + b)

= A(A⁻¹x + b) = x + Ab

= TAb(x)

(15)

+ ä

P 1.11

A³_ affine ¨8_ ^|9½+Ë A³(R)Ér6£§`¦ëß7á¤ôÇ.

(1) ∀F1, F2∈ A³(R), F2F1∈ A³(R).

(2) ½Ó1px¨8F₀= T₀L_Is >rFôÇ. (0Ér%ò7', IH½Ó1px'§>=) (3) ∀F ∈ A³(R), F⁻¹∈ A³(R)s >rFôÇ. (&ño 1.10\ _ #) (4) ∀F1, F2, F3∈ A³(R), F3(F2F1) = (F3F2)F1

û B' å

ç

ßéßy [O"î

(16)

+ ä

P 1.12

A³\"f 1 ¨8çH GL(3, R)õ ¨î's1lxçH T³(R)Ér affine ¨8çH A³(R)_ ÂÒìrçHs.

(17)

+ ä

q

A²_ affine ¨8\"fª_ ²ú©6£§¨8:







x⁰₁= ax₁− bx₂+ c x⁰₂= bx₁+ ax₂+ d

,

a −b b a

> 0

A²_ affine ¨8\"f6£§_ ²ú©6£§¨8:







x⁰₁= ax1+ bx2+ c x⁰₂= bx1− ax2+ d

,

a b b −a

< 0

(18)

affine ¨î A²_ ²ú©6£§¨8\ @/ #

P → P⁰, Q → Q⁰ s

,

P⁰Q⁰= kP Q

`

û B' å

P = [x₁, x₂], Q = [y₁, y₂], P⁰ = [x⁰₁, x⁰₂], Q⁰ = [y⁰₁, y⁰₂]s , P Q²= (x1− y1)²+ (x2− y2)²

2 2

(19)

A²\Hy_ >h¥Æs \Ol M:ëH\, &ño 1.14\"fHÄ»9þto×¼

¨î

+ ä

P 1.14

û B' å

¿ º f`¦







u : u₁x₁+ u₂x₂+ u₃= 0 (u₁, u₂) 6= (0, 0)

v : v₁x₁+ v₂x₂+ v₃= 0 (v₁, v₂) 6= (0, 0) .

u ⊥ v{9 ¸| Éru1v1+ u2v2= 0.

(20)



u⁰: u⁰₁x⁰₁+ u⁰₂x⁰₂+ u⁰₃= 0

v⁰: v⁰₁x⁰₁+ v₂⁰x⁰₂+ v⁰₃= 0







u : u⁰₁(ax₁− bx₂+ c) + u⁰₂(bx₁+ ax₂+ d) + u⁰₃= 0 v : v₁⁰(ax₁− bx₂+ c) + v⁰₂(bx₁+ ax₂+ d) + v⁰₃= 0

, 7£¤,







u : ( )x1+ ( )x2+ = 0

v : ( )x1+ ( )x2+ = 0

.

"f,







u₁= u⁰₁a + u⁰₂b v₁= v⁰₁a + v₂⁰b

,







u₂= u⁰₂a − u⁰₁b v₂= v₂⁰a − v₁⁰b

.

0 = u₁v₁+ u₂v₂= (a²+ b²)(u⁰₁v₁⁰ + u⁰₂v⁰₂)s¦ a²+ b²6= 0sÙ¼Ð

(21)

+ ä

P 1.15

² ú

ÒìrçHs.

û B' å Ã > V

(1) ª×ª → ª

(2) ª×6£§→ 6£§, 6£§×ª → 6£§ (3) 6£§×6£§→ª

(22)

ª_ ²ú©6£§¨8S :







x⁰₁= x₁− 2x2+ 3 x⁰₂= 2x₁+ x₂+ 5

\

¦Òqty .

Ô

j &h A = [0, 0], B = [1, 0], C = [0, 1]\ @/ # 4ABC −→ 4A^S ⁰B⁰C⁰s

A⁰=, B⁰=, C⁰=

(23)

¹M 5

6

£

§_ ²ú©6£§¨8T :







x⁰₁= 2x1+ x2+ 1 x⁰₂= x1− 2x2+ 2

H

ª

_ ²ú©6£§¨8S :







x⁰₁= 2x1− 2x2+ 1 x⁰₂= x1+ 2x2+ 2 ü<

x1»¡¤\'aôÇ ìøÍ R : x⁰1= x1, x⁰₂= −x2_ YL SR, 7£¤,T = SRs.

[

j &h A = [0, 0], B = [1, 0], C = [0, 1]\ @/ # 4ABC −→ 4A^T ⁰B⁰C⁰s

A⁰=, B⁰=, C⁰=

§F 26Aá¤ÕªaË> ÃÐ¸. sQôÇ T \¦ìøÍSX@/ ôÇ.

(24)

a :

@ : @' Ö << K 1.3

1∼3 : /BN:xõ]jÓüt 4∼6 : ¸Z> õ]jÓüt(µ1Ï³ð)

< K1 ç ¡ AFFINE M  È Ñ