高等几何

21 Æ ! !"#$%&'()*+, ./0123456...

Author: 周兴和

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21

Æ

! !"#$%&'()*+, ./012345678, 9 :;<=>?9!@"A#BCDE5FG, H$ %&'() -5I(-5J*HB+KL ,FG, 20

-./012 34

56IM, NOP-4QR ST %&J U, V,5 W, & J X, 7YQ Z '[\8 ] ^ 9:; ? JKLMPNOP (CIP)

/

—

2003.9

21 ISBN 7-03-011617-8 ···

···

-

CIP

(2003)

张克忠 /

-

051897

/

4Q f _`abcd_e QRghi ST jUklmnV opWqXrsY tt Zu [ l vwx Zuyz\] \_x abP c\^x `_xdef {+Sgh|ijÆR}k~! l 16

100717

http://www.sciencep.com

2003 2005

9 6

*

B5 (720×1000) 11 1/2 214 000

7 001—9 001

20.00

(

(

)

)

O18

20

Klein

ii

1999 2000

5 2003

§1.1

§1.2 §1.3 §1.4 §1.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Desargues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

§2.1

§2.2 §2.3 §2.4 §2.5 §2.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

§3.1

§3.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

iv

§3.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

§4.1 §4.2 §4.3 §4.4 §4.5 §4.6 §4.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 Pascal Brianchon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Pappus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Gauss, Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Cantor Poincar´e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

§5.1 Euclid §5.2 §5.3 §5.4 §5.5 §5.6

§5.7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Desargues !" # " " ! " 1. 1.1 l, l O " l, l $! l l . " O l P OP l P , P P l O ( 1.1). OP . §1.1

! 1.1

P l P l , P P . P P l " l l X, #

!% #" X = l × l l l X. X , . " # l U , OU//l , $ OU l Æ U l U l . $ l ! V . $ ! $ . 2. 1.2 π, π O $! π π . π P , " OP π P , P P π O . OP ( 1.2). 2

1)

! 1.2

P P , P P . $ π × π = x( x π π ), x X , x . " O # π u, U O OU π , U ! u π . $ π ! v . $! $

! !! # $ Æ

%$ &% $ &% 1) &$ (injection) '&$ (surjection) ($&'$ (bijection), %&&&'

' 3 )( !

(i) % (ii) % % " 1.1 (1) $ ! #. (2) $ "& $ "& (3) $ "& ! §1.1

#

' $ ! %. P "& $ ! %. l "& "& "& !. "& % $ "& &% $

& $ "& $ "&

1.3 %"& &; $ "& "& & ! ; $ "& & ! . 1.1 (1) ! (); (2) ! (). )* " #( 1.1 ) %) % "

&& !'( &"#) .

∞

∞

!% #"

4

" (1) 1.3 *$ * $ $$ "& % % $ !$$ (*) $ 1.

! 1.3

(2) $%&*. %+* #Æ #Æ ' $!$) % ( 1.4, C, D A, B. 1.5, C, D A, B.

! 1.4

! 1.5

$ A, B # ÆÆ &

C, D A, B #Æ C, D A, B #Æ $ 2. $$ "&

' 5 " (1) # '+ (Æ'+ , +!!); # '+ ( 1.6). §1.1

! 1.6

! 1.7

# '+ # '+ 1.7, l , l #"#$%Æ ) "#$%'+ $ (2) $ $!) 1.8, )!! π *) O " ) ( ") #" 1

! 1.8

2

!% #" & ) * ) $!) $ $ ( ). $$ $ " " $ $! #&)$$ ( 1.9), %'& ) $ #"& &) "& $ 6

! 1.9 $!!* $$ &)$'!" ,, * !! * $ ! 1.10 * $ A, A; B, B ABAB # '!", ( 1.11), "- M¨obius %

! 1.10

! 1.11

' 7 !+ - (1) '. (1) ,. (, ( ,, ( !. ( !. l & l(A, L (( L(a, B, C, · · · ) ! l(P ). b, c, · · · ) ! L(p). $ $ % %"&

. ( ) (&) $ &% %%&%$( &% & (2) ). (2) ). " (, " (, " !. " !.

"!" " π !" π

- !+, - !+. ' &%# (3) +. (3) +. " " ' ! ! ' ' ! ABC abc !( §1.1

. /

1.1

/ ϕ * π ( π ) (#"(!) f * π +)"*)) π +!) p, q . f $* V , p, q / π +)+%** p , q . ,"+ p //q .

1.

!% #" 2. ,!& (#"() π +)-!0,,+#* π +)!&-0,,) 3. / ϕ * π ( π ) (#"(!) f * π +)"*)( P, Q *!) f + )&+*( R * π ##/ f +)-!*),"+ ∠P RQ / π +)+*+#) 4. .') π +)!,+!) p - O *+# (+#( π +,!) p . ,"+O -.1(!) p "(!&+*) 5. ./ !,"( (#"$))/.!,%*23!. (1) 0//*,/ (2) !*/*,/ (3) -,/ (4) 0,,/ (5) 0/ (6) 1%0)/ (7) %",,/ (8) ,-!)/ (9) ,$!!)/ (10) * *)4) 6. 5 . $0!,/ (#"/2#-) (&-*/%!,) . 8

12&& * 0 $ &%( %3 #++( —— ('3 %"& #+ % 3 #4 ( 3 n-01+, R R n Æ §1.2

n

Rn = {(x1 , x2 , · · · , xn )|xi ∈ R},

0 ∈ R n $ R \ {0} ∼: x, y ∈ R \ {0}. x y $ ∼ %% ρ ∈ R(ρ = 0) x = ρy. ∼ R \ {0} "% n

n

n

n

RP n−1 = (Rn \ {0})/ ∼

(n 2).

RP n $ x ∈ RP RP Æ R x = (x , x , · · · , x ) &Æ n x. x ' " n−1

n−1

1

2

n

n−1

n

'+)'% -1 9 RP '*% , %Æ x = [x , x , · · · , x ]. $ n §1.2

n−1

1

2

n

(RP n−1 )∗ = (Rn \ {0})/ ∼

(n 2),

! R % ) RP (RP ) (

.2/ 1. ( ) 1.4 % P , 3 x, x /x = x( x = 0) *( (x , x ) P .2/; "& P , (x , 0)(x = 0)

3 0.2/. 1.4, "& ( Æ" (x , x ) P ρ = 0, (ρx , ρx ) P ( (0, 1), "& ( (1, 0). 1.4 #+ % 1.4 RP $ ϕ, ϕ # 2. ( ) P l , l ( ! l , l && ), " P & l ∦ l ; ""& P , l //l . , ( n

n−1

n−1 ∗

1

2

1

1

2

2

∞

1

1

1

2

2

1

1

1

2

x=

%

B1

C1

B2

C2

A1

B1

A2

B2

1

1

2

i = 1, 2.

& P 3 P (x, y),

∦ l2 , P C1 C2 , y = A1 A2

B1 x1 = B2

1

∞

li : Ai x + Bi y + Ci = 0,

" l

2

C1 , C2

A1 A2 . B1 B2

C1 x2 = C2

A1 A2

A1 , A2

B1 = 0 ( B2

l

A1 x3 = A2

1

B1 , B2

∦ l2 ).

2

!% #" x = x /x , y = x /x , Æ x : y : 1 = x : x : x . (x, y, 1) *( (x , x , x ) P " l //l , "& P , % P l , l "& x = A B = 0. # & (x , x , 0) *( A B P $ l = l x , x $) x = 0, ! x /x A A A B = = , = = . B B A B A A = 0(l ∦ x )) ! B B = 0(l ∦ y )), ) B B = 0, 10

1

3

2

3

1

1

1

2

∞

1

1

2

2

1

1

2

∞

1

∞

1

3

3

2

3

2

2

2

1

2

1

1

2

2

2

2

2

1

1

1

2

2

1

i

1

i

1

C2 A2 C1 A2 C1 A1 C2 − − x2 A2 C1 − A1 C2 B2 B1 B2 B1 B2 B1 B2 = = =− B2 C1 C2 B1 C2 C1 x1 B1 C2 − B2 C1 − − B1 B2 B1 B2 B1 B2

2

=−

A2 A1 =− . B2 B1

x /x l 1' (x , x , 0) λ = x /x "& (x , x , 0) 0 (1, λ, 0). λ = 0(Æ x = 0) x )" & y )"& (0, x , 0), (x = 0). ( 1.5 % P , "3( (x, y), *( (x , x , x ) .2/, x = 0 % x /x = x, x /x = y. y )"& (x , x , 0), x = 0 % x /x %1' y )"& (0, x , 0), 2

1

1

i

1

2

2

2

1

2

2

1

2

3

3

1

1

2

2

3

2

3

1

2

2

1

%3 0.2/. $ 1.5, "& ( "& ( Æ" (x , x , x ) P ρ = 0, (ρx , ρx , ρx ) P # P P (ρx , ρx , ρx )(ρ = 0), P (x , x , x ). $ (0, 0, 0) (0, 0, 1), x )"& y ) "& (1, 0, 0) (0, 1, 0). % 1.5 #+ "& 1.5 #+ %1' 1.5 RP $ x2 = 0.

1

1

2

1

2

3

3

2

3

1

2

3

2

'+)'% -1 11 ϕ, ϕ # x, y, a, b, · · · (x , x , x ), (y , y , y ), (a , a , a ), (b , b , b ), · · · . 1 1.1 (1) §1.2

1

2

P1 (0, 0),

3

1

P2 (1, 0),

2

3

1

P3 (0, 1),

2

3

P4 2,

1

5 3

2

3

.

3x − 4y + 1 = 0 "& 2 (1)

P (0, 0, x ), x = 0, P (0, 0, 1). P (ρ, 0, ρ), ρ = 0, P (1, 0, 1). P (0, ρ = 0, P (0, 1, 1). ρ, ρ), 5 P 2ρ, ρ, ρ , ρ = 0, P (6, 5, 3). 3 (2) (1') λ = 3/4, "& 3 1, 4 , 0 , ! (4, 3, 0).

.2/34

1.2 (2) 1

3

3

1

2

2

3

3

4

4

3

ui xi = 0.

(1.1)

(1.1) 5 (1.1) .34. 35 % l, 3( i=1

l : Ax + By + C = 0 (A2 + B 2 = 0),

(1)

x = x /x , y = x /x ( 1

3

2

3

x1 x2 +B + C = 0, x3 x3

(2)

Ax1 + Bx2 + Cx3 = 0.

(3)

A

Æ

(3) Æ (1) $ "& (B, −A, 0) 5 (1.1). "& l , * x = 0, Æ (1.1) u = u = 0, u = 0 (1.1) " u + u = 0, $ u x + u y + u = 0 $ "& (u , −u , 0) " u = u = 0, u = 0, Æ x = 0, "& ∞

3

2 1

2

3

3

2

1

1

2

3

2 2

1

1

2

3

12

4

!% #" 1.1

u1 x1 + u2 x2 = 0.

$ & "

.2/ + &% 2 , &%

(,3(), Æ

. ( u1 x1 + u2 x2 + u3 x3 = 0,

% ρ = 0, ρu1 x1 + ρu2 x2 + ρu3 x3 = 0

2(!$ ($ 1. 1.6 l u x + u x + u x = 0. ( l .2/, [u , u , u ]. $ 1.6 !

(1) & $*( [u , u , u ] $*( [u , u , u ], ! u x = 0. ρ = 0, [ρu , ρu , ρu ] ≡ [u , u , u ], % [0, 0, 0] (2) "& u x = 0(u = 0) ! x = 0, [0, 0, u ] ! [0, 0, 1]; u x + u x = 0, [u , u , 0](u , u $); y ) x = 0, [1, 0, 0]; x ) x = 0, [0, 1, 0]. &' '%*% $ (RP ) $ ψ, ψ # 1 1

1

2

2 2

3 3

3

1

1

1

2

3

2

1

2

2 ∗

3

3

1

3 3

1 1

2

2 2

2

3

i=1

i i

3

3

3

3

1

2

1

2

'+)'% -1 13 - 3( 15 65.2/. $ # ψ $ # ϕ ! 1 u,v, · · · 2 [u , u , u ], [v , v , v ], · · · . 2. 1.3 x u ⇐⇒ §1.2

1

3

2

3

1

2

3

ui xi = 0.

(1.2)

,$ " %% ! .34 i=1

1.7

[u1 , u2 , u3 ]

P

P

.

1.4

a = (a1 , a2 , a3 )

a1 u1 + a2 u2 + a3 u3 = 0.

(1.3)

$) [u , u , u ] " 35 1.3, u = [u , u , u ] a = (a , a , a ) (& a u + a u + a u = 0. 1.7, (1.3) a $) " (1.3), a , a , a $" (1.3) # (1.3) a = (a , a , a ). $ 1.3 1.4 x 1

2

3

1

1 1

2 2

2

3

1

2

3

3 3

1

1

2

2

3

3

3

xi ui = 0.

(1.4)

!$ x *# x u ) 5 5 ' u x ) * (1.2) 5 !- $ $" a = (a , a , a ), (1.3) 5 a u ) Æ 1 1.2

(1) (3, 1, 5); (2) (3) x )"& (4) x + 2x + x = 0 "& 2 (1) 3u + u + 5u = 0; i=1

i

i

i

i

1

i

i

1

1

2

3

2

3

2

3

!% #" (2) (0, 0, 1), u = 0; (3) x )"& (1, 0, 0), # u = 0; 1 (4) (1') λ = − , "& 2 (1, λ, 0), Æ (2, −1, 0), 2u − u = 0. % x = 0, x : x , Æ x + 2x + x = 0 "& (2, −1, 0), 0 "& 5 . 0.2/ 1.8 u = [u , u , u ], " u = 0, [U, V ], 14

3

1

1

3

1

1

U=

2

2

3

u1 , u3

2

1

2

3

3

V =

u2 . u3

$ 1.8 [u , u , 0] 1

2

U x + V y + 1 = 0.

(1.5)

"& &

$%Æ"& 40 67 " (x , y ) x U + y V + 1 = 0; " [U , V ] "& U x + V y + 1 = 0. ) "& & $ 2' 1 & + ) ,3( 5(& &, (Æ3() + ) , "+ 3( (Æ, *). 7&6.2/ !8 - 0

0

0

0

0

0

0

0

§1.2

'+)'% -1 1.5 a, b ⇐⇒ 

/



a1

a2

a3

b1

b2

b3

/

 = 1.

35

15

1.5  a  1 b1

a, b ⇐⇒ 

a2

a3

b2

b3

 = 1.

0 1.5. a, b ⇐⇒ a, b RP &% &% ⇐⇒ 1/2 2, a, b ∈ (R \ {0}), # 1/2 1, / 1. 1.6 a, b 1.6 a, b 2

3

 a  2 b2

35

x1 a1 b1

x2 a2 b2

a3 a3 , b3 b3

x3 a3 = 0. b3 a1 a1 , b1 b1

 a2  . b2

 a  2 b2

0 1.6. a, b u x

1 1

  u x + u2 x2 + u3 x3 = 0    1 1 u 1 a1 + u 2 a2 + u 3 a3 = 0     u b +u b +u b =0 1 1

2 2

u1 a1 b1

2

a2 b2

a3 a3 , b3 b3

u3 a3 = 0. b3

a1 a1 , b1 b1

 a2  . b2

+ u2 x2 + u3 x3 = 0 ⇐⇒

u , u , u $ 1

u2

3

3 3

x1 ⇐⇒ a1 b1

x2 a2 b2

x3 a3 = 0. b3

( $ #- 1.7 a, b, c 1.7 a, b, c

⇐⇒

 

 a1

a2

1

b2

c1

c2

/  b

35

a3

  b3  = 2.  c3

⇐⇒

 

 a1

a2

1

b2

c1

c2

/  b

a3

  b3  = 2.  c3

1.6, /2 3 # / 2.

!% #" 1.8 a, b 1.8 a, b & %% (( la + mb(l + m = 0). %% la + mb(l + m = 0). 35 0 1.8. c = la + mb 16

2

2

2

a1 b1 c1

a2 b2 c2

2

a3 b3 = 0. c3

# c a, b & c a, b & a1 b1 c1

a2 b2 c2

a3 b3 = 0. c3

Æ a, b # a, b " c = la + mb, % l + m = 0. $ 1.8, c = la + mb a, b . m 798. " λ = l , 0.798 c = a + λb. x x = ∞(x ∈ R, x = 0), = 0(x ∈ R, x = 0), l = 0 λ = ∞, %2 0 ∞ ∞ = 1. R = R ∪ {∞} R &07:, $ c = a + λb ∞ R $ $ λ = 0 c = a; λ = 1 c = a + b; λ = ∞ c = b. * $ 1.8 $(**( * $ 1.9 a, b, c 1.9 a, b, c ⇐⇒ p, q, r(pqr = 0), ⇐⇒ p, q, r(pqr = 0), 2

2

pa + qb + rc = 0.

pa + qb + rc = 0.

- $ $ -5 2() 0$ 5

'+)'% -1 1 1.3 a = (3, 1, 1), b = (7, 5, 1), c = (6, 4, 1). (1) 30 (2) l, m, c la + mb 5 λ, c = a + λb. 2 (1) . (2) c = la + mb. Æ

§1.2

6

7

3













17

            l 1  + m 5  =  4 ,       1 1 1

l = 41 , m = 34 , Æ c=

3 1 a + b. 4 4

6# a, b, c c = a + 3b, Æ λ = 3. %       6 3 7           ρ 4  =  1  + λ 5      1 1 1

  . 

λ = 3. # a, b, c c = a + 3b. 1 1.4 (1) a = (a , a , a ), b = (b , b , b ), c = (c , c , c ) 0

!

ρ,

1

2

3

1

lai + mbi + nci

2

3

1

2

3

(i = 1, 2, 3),

l, m, n $ (2) d = (d , d , d ) % a, b, c, d " 0 $ p, q, r, s, 1

2

3

pai + qbi + rci + sdi = 0

(i = 1, 2, 3).

$26#

(a , a , a ), (b , b , b ), (c , c , c ), (d , d , d ), 1

2

2

3

1

2

3

1

2

3

1

3

35

ai + bi + ci + di = 0

(1)

(i = 1, 2, 3).

d 2 d a, b, c

18

$

!% #" 1. d a, b, c 5% l, m, n

2. d a, b, c ( d & 5 l, m, n $ 3. d "Æ a, b, c, d t a, d b, c 1.8 di = lai + f ti

(i = 1, 2, 3).

d = a, t, lf = 0. ti = gbi + hci

(i = 1, 2, 3),

gh = 0. - 5 di = lai + mbi + nci (2)

(i = 1, 2, 3).

$ (1) pai + qbi + rci + sdi = 0,

a, b, c, d "# pqrs = 0. a c = rc , d = sd , ( i

i

i

i

= pai , bi = qbi ,

i

ai + bi + ci + di = 0

(i = 1, 2, 3)

1.4 )" *

, +61 $ " & ( 3 A , A , A 23 x, y )"& 2( (1, 0, 0), (0, 1, 0), (0, 0, 1), x 1

2

3

x = (x1 , x2 , x3 ) = x1 (1, 0, 0) + x2 (0, 1, 0) + x3 (0, 0, 1).

A , A ! A + 52 x, 7 I, % 1

2

3

I = (1, 1, 1) = (1, 0, 0) + (0, 1, 0) + (0, 0, 1).

'+)'% -1 19 A , A , A , I (A A A |I) 65.2 /*, A A A 2/ +, I ;9. 6 I ' # 1 1.5 A A A 3 §1.2

1

2

3

1

1

2

3

1

2

2

3

3

P1 (p, g, h), P2 (f, q, h), P3 (f, g, r).

0 P P A A P P A A P P A A (2) " f gh = pqr, 0 A P , A P , A P A P , A P , A P 35 A , A , A A (1, 0, 0), A (0, 1, 0), A (0, 0, 1). (1) 1.6, P P (1)

2 3

2

3

3 1

1

1

2

2

2 3

3

1

3 1

3

1 2

1 3

1

2 1

2

1

2

3 2

3

2 3

x1 f f

Æ A A 2

x2 q g

x3 h = 0, r

(qr − gh)x1 + (f h − f r)x2 + (f g − f q)x3 = 0.

3

x1 = 0.

1.6 , P P A A (0, g − q, r − h). P P A A (p − f, 0, h − r). P P A A (f − p, q − g, 0). 30

2 3

3 1

2

3

3

1

0 p−f f −p

1 2

g−q 0

q−g

r−h h − r = 0, 0

# 1.7, x1 p−f f −p

Æ

x2 0 q−g

h − r = 0, 0 x3

x2 x3 x1 + + = 0. p−f q−g r−h

1

2

20

A P hx − qx = 0; A P gx − px = 0. ⇐⇒

(2) A3 P1

1 2

1

2

2

h 0 −r 0 g −p

3

2 3

!% #" f x − rx = 0; 3

1

−q f = 0, 0

Æ f gh = pqr. 0 A P , A P , A P (& f gh = pqr. 1 1.6 ABC ! BC, CA, AB α = 0, β = 0, γ = 0, A, B, C 0 ⇐⇒ qβ−rγ = 0, rγ − pα = 0, pα − qβ = 0, p, q, r 35 ⇐= =⇒ ) ! A qβ − rγ = 0. B 6# p, rγ − pα = 0. C 6# q , pα − q β = 0. %! l, m, n(lmn = 0) 1 3

2 1

3 2

l(qβ − rγ) + m(rγ − pα) + n(pα − q γ) = 0,

Æ

(np − mp)α + (lq − nq )β + (nr − lr)γ = 0.

α = 0, β = 0, γ = 0 Æ q = q .

. /

1.

1.2

,/ -*)'% - 1(34+4324-4!2)

(1)

(2, 0),

(1, 3),

(4, 1),

5 ,2 ; 3

- 34 *.5)635*/ (3) !) 3x + y = 0 +)635*) 2. .',/(,/ -*)/'% - 1) √ √ P (2, 4, −1), Q( 10, − 6, 2), R(0, 1, 0), S(0, 4, 3), T (1, 4, 0), U (1, 0, 4). 3. .'%0-(--544(. (±1, ±1, ±1) (-78/&76)*.4+5,-,+, 7.)/'%-1) 4. ,/ -!))'% - 1) (1) x 3/ (2) y 3/ (3) 635!)/ (4) x − 2x − 3x = 0; (5) 6(6*(8-* 2 )!)/ (6) * (0, 1, 0) . (1, 0, 1) ).)) (2)

1

2

3

'+)'% -1 5. .',/(, 4 . -!))/'%) - 1) 6. ,/ -!)+)635*)

§1.2

(1)

7.

x1 + x2 − 4x3 = 0;

(2)

x1 + 2x2 = 0;

x2 − 3x3 = 0;

(4)

x1 + 5x3 = 0.

,/ -)-1478)!)..)

(3)

21

(1)

[0, 1, 1];

(2)

[1, 1, −1];

(3)

[1, 0, 1];

(4)

[1, −1, 0];

(5)

[2, −1, 5];

(6)

[0, 0, 1].

8. (1)

u1 = 0;

(2)

u3 = 0;

(3)

u2 − u3 = 0;

(4)

2u1 + u2 = 0;

(5)

u1 + u2 + 2u3 = 0;

(6)

u21 − 5u1 u2 + 4u22 = 0;

u21 − 3u1 u2 + 2u23 = 0;

(8)

au1 + bu2 = 0.

/ )-1..-783!!,.

,/ -*)'%) -1..) (1) x 3+)635*/ (2) y 3+)635*/ 1 )!)+)635*/ (3) 8-* − 2 (7)

9.

(4)

(0, 0, 1);

(5)

(2, 4, −3).

.',/(,+. -*)/'%)- 1..) 11. 9,- 2x + x + x = 0, 3x − 4x + 2x = 0, 4x + x − 3x = 0 *,)/*,) 8*-1) 12. 9,- A(1, 2, 3), B(2, 2, 1), C(3, 4, 3) *8*)/),)/,..) 13. ,/ !))0* - 1...) (1) x + 2x − 4x = 0 . 2x − x + x = 0; (2) x − 2x = 0 . 4x − 5x + x = 0. 14. 74/* P (1, 4, −3), P (0, 2, 5), P (3, 8, −19). (1) "-+ P , P , P /*$)/ (2) ,4/!)../ (3) 44 P )'% - 1:, P = P + λP . 15. //*, A B C . A B C //!#( B C . B C 0 X, C A . C A 0 Y , A B . A B 0 Z, 95 X, Y, Z /*$)),"+/!) A A , B B , C C $ *) (78+//*, A B C . A B C ), B C , C A , A B . B C , C A , A B ) '%..8%* α = 0,β = 0,γ = 0; . α = 0,β = 0,γ = 0, 4/ X, Y, Z 4/!)'% ..* δ = 0, 6)07814;)) 16. / O(f, g, h), A A A * - 1/*,( A O, A O, A O %*. A A , A A , A A 0 P, Q, R; QR, RP, P Q %*. A A , A A , A A 0 L, M, N . ,"+ L, M, N /*$ )(-,+/!))..) 10.

1

1

2

1

2

2

3

1

3

1

1

1

1

2

2

3

2

1

1

2

2

1

2

3

3

3

3

1

3

3

2

3

1

2

1

2

2

1

2

2

1

1

2

2

1

1

1

1

1

2

1

1

2

2

1

1

2

1

2

2

3

3

3

1

1

1

1

1

2

2

1

1

1

2

2

2

2

1

2

2

2

2

2

1

2

1

2

2

2

3

2

3

3

1

1

2

!% #" 17. / O *+!) x − kx = 0 +).*( A A A *:<'% - 19) - 1/*,) A O, A O %*. A A , A A 0 Q, R. ,"+ QR . A A 0+* X. -,+!) A X )..) 18. / a, b, c *#$)/*)"-+#76/* l a+m b+n c (i = 1, 2, 3, l +m +n = 0) $))09,2* 22

2

2

3

1

3

3

1

1

2

3

2

1

i

l1 l2 l3

m1 m2 m3

3

1

i

i

2

2

2

n1 n2 = 0. n3

,!) ax + 2hxy + by = 0 +)635*)..) 20. ,"+.. u + 3u u − u = 0 4)+)635*/7/$!).5+) 21. 81/7+ < 9 1 2 . 3 2.3 . 3 2.3 8; :795; 08;5; λ = a /a 1<8:79; y 6 0 (0,1) =1& 2 5; 22. ,* 3u + 4u − 11u = 0 . 5u − 3u + u = 0 .)) - 1...) 23. ,6(!) [1, 1, 1], [2, 1, 3] )0*.* 2u + 3u + u = 0 )!)) - 1...( -,,+635*) - 1) 2

19.

2

2 1

2

2 2

1 2

1

1

2

3

1

2

3

1

2

3

9 : +2" Æ "+$ − 3 §1.3

0 -0=> (

)

Æ )"&% ) " 1.2 (1) P l )"

#" P l l P P l l % P . (2) P l )"

l P P l P l l % P . 1.9 π P L P &% π . L &% π . % ? P $ §1.3

23

ψ : L −→ (RP 2 )∗

ϕ : P −→ RP 2 ,

P ∈ P l ∈ L, " ϕ(P ) = x,

ψ(l) = u

P l ⇐⇒ u x + u x + u x = 0. π P L 0 ! -0=>, π = (P, L). P $ (ϕ, ψ) π 2/:. ϕ ψ 2/: 2/:, # 2/. ! 1 x = (x , x , x ), u = [u , u , u ], ! x, u. $ P, 1.10 1 1

2 2

3 3

1

1

2

2

3

3

u1 x1 + u2 x2 + u3 x3 = 0.

(1.6)

! x ) u (1.6) [u , u , u ]; (x , x , x ). $ 1.10 P # ψ $ # ϕ 15

0#) (0;)

" " π & π π 0;, ! #).

$ 1 1.7 $ i

1

2

3

i

1

2

3

!% #" P = {}, L = {}. P # (ϕ, ψ) # %"& % "& '$3 153 # $ $ 93' - $ 1 1.8 *$ (* ): * π = (RP , 24

2

R

(RP 2 )∗ ).

P = RP , L = (RP ) . x u %% 2

2 ∗

u1 x1 + u2 x2 + u3 x3 = 0.

π # (α, β), R

α : RP 2 −→ RP 2 ;

β : (RP 2 )∗ −→ (RP 2 )∗

&2# ! π = (P, L), π P l % %*$ π ϕ(P ) ψ(l) ) *$%% #$*$ 93' $ $ #%=) $*$- 4-$ ' $*$%1 $ (+ ). % $ "4& # $ &%$4 1 1.9 π = (P, L), "+ π = (P , L ), P = L, L = P. π π l π P %% π π P π l %" π # (ϕ, ψ), π # (ψ, ϕ). π = (P , L ) π = (P, L) @. (π ) = π. R

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗ ∗

∗

∗

#" 25 1 1.10 $ 0!)! #&)!! * $ # " $ # ! * $

2/ $ $ §1.2 3 (A A A |I), % # $ # 22 # # 1.10 A (1, 0, 0), A (0, 1, 0), A (0, 0, 1), I(1, 1, 1), " # A , A , A , I 5 §1.3

1

1

1

2

2

2

3

3

3

ρI = A1 + A2 + A3

(0 = ρ ∈ R)

(1.7)

<2/*, (A A A |I). A A A 2/ +, I ;9. ! $ Æ 3 1.11 P, Q, R, E,

(1) P, Q, R, E (2) " #

P (p , p , p ), Q(q , q , q ), R(r , r , r ), E(e , e , e ) 1

1

2

2

3

3

1

1

2

2

3

1

2

3

1

2

3

3

ρ(e1 , e2 , e3 ) = (p1 , p2 , p3 ) + (q1 , q2 , q3 ) + (r1 , r2 , r3 ). (0 = ρ ∈ R)

(1.8)

" P QR 2/ +, E ;9 2/*,

35

(P QR|E).

0 X, (P QR|E) ! X # X (x , x , x ). $ ( x , x , x , ρ(x , x , x ) = x (p , p , p ) + x (q , q , q ) + x (r , r , r ), 0 = ρ ∈ R, 1

2

∗ 1

∗ 1

∗ 2

∗ 3

2

1

2

3

∗ 2

∗ 3

1

1

2

3

3

3

1

2

3

!% #"

26

Æ



x∗1





 p1

      ρ  x∗2  =  p2    x∗3 p3

q1 q2 q3

 x1

r1

      r2 x .  2  r3 x3

0 = ρ ∈ R

(1.9)

$ P, Q, R (1.9) 1/# $ ρ (x , x , x ) !$$ ρ (x , x , x ) RP (P QR|E) %( (1.9) ! X # (x , x , x ) X (P QR|E) .2/, 1

1

1

2

2

3

2

3

2

3

$ 1.11 P, Q, R, E (P QR|E) 2 (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1). $ * '

' (1.9) 55 $ P, Q, R / (1.9) 50505 / (1.9) 05 2/ . () (1.9) 5 + 5' (x ) (x ) + (1.9) 55' X(x1 , x2 , x3 ).

i





x1

i





p1

      ρ  x2  =  p2    x3 p3

q1 q2 q3

r1

x1



      r2 x ,  2  r3 x3

0 = ρ ∈ R

.

(1.10)

' (1.10) 5 2/ A=. 05 P (p ), Q(q ), R(r ), % 6 E(p + q + r ). ' $# & (1.10) 5

0 (-0=>)

$" 0 RP $% (3 15 ! $ 15 Æ i

1

i

i

i

i

i

#"

§1.3

27

P, Q, E, # 3 P (p , p , p ), Q(q , q , q ), E(e , e , e ) 1

2

1.11

3

1

2

3

1

2

3

ρ(e1 , e2 , e3 ) = (p1 , p2 , p3 ) + (q1 , q2 , q3 ). (0 = ρ ∈ R)

-2/*, (P Q|E). P, Q

;9

.

(1.11) ,E

X(x , x , x ), $ λ, µ ρ(x , x , x ) = λ(p , p , p ) + µ(q , q , q ), 0 = ρ ∈ R, (1.12) λ, µ X (P Q|E) .2/, (λ, µ). 1.11, P, Q, E (P Q|E) (1, 0), (0, 1) (1, 1). 3 ' ' 3 * λ . P, Q, E (P Q|E) ∞, 0, 1. µ 0 – B 1 :!: :& % C 3+ 1.9 , CP (CP ) , B % 3+> §1.1, 3 3 $ 3 %% :& % 2+) :( 3 3 $ : :.5 (/ ) (:&% 0 – B. $ – 3 : % : : 65$ ( (i, i, i) 1.12 – 3 P (x , x , x ) 0, , 0 = ρ ∈ C, ρx ∈ R , P >. > :: :: 1.13 &% (!)a = (a , a , a ), b = (b , b , b ), "3 1

1

2

3

1

2

3

2

3

1

2

3

2

2 ∗

1

2

3

j

1

2

3

1

2

3

!% #"

28

ρ = 0, ρbi = a¯i ,

&% a b C?B !. b = a¯ ! a = ¯b. $%1( ::Æ2&%, Æ – 3 2 (1.2) 5# -2 – 3 - (1) x u (& (1) u x (& x¯ u¯ u¯ x¯. (2) : x u ( (2) : u x ( & x¯ u & u¯ x. (3) !! (3) %! Æ : ! Æ : (4) Æ : (4) Æ : (5) :%% (5) :%% ' "? – 3 .3> 1 1.11 0 (1+i,−1+i),(1,1+i), (i,−1−i) 3 "30

1 + i −1 + i 1 1 1+i 1 =0 i −1 − i 1

Æ

2(1 + i)x1 − (1 − i)x2 − 2ix3 = 0,

2(1 − i)x1 − (1 + i)x2 + 2ix3 = 0.

Æ (1, 2, 2).

#" 29 7@+'( &% 2%#)$ ):6* ( ') ( +). $ $ ) :6* 46;'+ ("" 6( !*(* §1.3

. /

1.3

1. @#")43?)-' α +)67)4.,@)4 D *;*<( α +)67 !).,+);!,@!) D *!)(5+!).*77Æ-57-8!)A;&)4)9 "-;2@#")!&3?) 2. @#")83?)//<4<-8 E 4+!* S. / π = {E ( S )<=!)} ∪ {E ( S )<=}, 9 π * E (* S )!)8.8(), )!)D * π );*<(, ) D* π ) ;!)<(-5+;*<.;!)<77Æ-57-/ E 7.)!)/7.)+)9" -;2@#")!&3?) 3. /#"+/* A, B, C /54+#" - 19/)'% - 18%* (0, 0, 1), (1, 2, 3), (2, −1, 4). "-//*#$)(2,!* E, 91A)#" - 19 (ABC|E), -,+#" - 1 -6)6B) 4. ,* A(1 + i, 2 − i, 1) . B(1 − i, 2 + i, 1) ).)..) 5. / a, b *@ – :#"+)&*)9<+/ 93+ (1) = a = 0, < (=$ E =3)b = 0, 2( b a a.b$ E ⇐⇒ = (i = 1, 2). b a (2) = a = 0, < (=$ E =3)b = 0, 2( b a a a.b$ E ⇐⇒ = * a = bb . a b 6. ,"+ A(2,i,1−i) . B(2+2i,1−i,2i) *!/$ E ;*(-,,.)..) 7. ,6(* (1,i,0) )@!)) 8. ,!) [2,i,3−4i] +)@*) 9. ,635!) x = 0 .+)0 (x − ax ) + (x − bx ) = R x )0*) 3

3

3

3

3

3

3

3

i

i

3

3

3

3

1

1

2

2

2

2

1

1

1

3

2

2

3

2

2 2 3

!% #" 10. /81@#")+< D ./B;+) RP )@C= R ;*-!3C= F , . D=(-,(= F +)/>/455.):7 F P (75/155);>(C:0?/3C= F +<-), ;=/(-+<3C= F +)#"(9*3C#")94 F *8C:3 3 /?.@03C#")>1)/=)D()7@#")-94;(?/83C# "+<-?:;9) 30 ∗

2

2

@7AB 0<<;# % @A 1. 1.14 @ !. 1.15 @BC. 1.16 &%2 @ §1.4

$ " Æ &%&% 22 + @@+ .

1.17

Σ.

Σ

Σ

Σ.

Σ

Σ

2 & ) (1) (2) ().

.

2.

).

(3)

" (

(2)

( (

(3)

" (

(1)

). ).

6 n n ( (4) 6 n n ( ") +" ") + n ' n " n ! ! n ' 6 n () 1.12 1.13 n = 3 n = 4 n (! n ), $ "6 n (6 n ) (4)

/86< 31 n (n ) 9* ! §1.4

! 1.12 >A/*,.>A/),

! 1.13 >A0*,.>A0), (5) n n ( (5) n n ( ") " ") " n n n n n(n − 1) ' 2 ! ! n(n2− 1) '

! 1.14 /*,./), n ! n

!% #" ( ") ! n ( "), $' n ! n n = 3, ( 1.14), @@+, '

' C% ' "463 DE " 1.15 1.16 F) &% 7 32

! 1.15 8<0*, DF +ABCD G A, B, C, D (4 ) H p, q; r, s; t, u

! 1.16 8<0), DF +abcd H a, b, c, d (4 ) G P, Q; R, S; T, U

(6

)

(3

)

) G !' P, Q; R, S; T, U (3 () G '

AC, BD; AB, CD; AD, BC

H '! ( H ! p, q; r, s; t, u X,

Y,

Z

H +

p × q; r × s; t × u

(3

)

a×c, b×d; a×b, c×d; a×d, b×c (6

x,

y,

z

G + P Q,

RS,

TU

(3

)

E3 2 1 1.12 + 1.17 2 25< !-", XY Z.

xyz.

/86< 33 =: ( 1.17). (Æ2). B@ &%&% 2 2 -,+ §1.4

! 1.17 $ (1) P, Q 2 l, a, b, c, d 5

(i) P, Q l (ii) a, b, l P , c, d, l Q.

p, q 2 L, A, B, C, D 5

(i) p, q % L (ii) A, B, L p, C, D, L (1)

q.

A, B, C, D, E 5 5 (2) a, b, c, d, e p, q, r, s, t, u, v 7 P, Q, R, S, T, U, V 7

(i) B, E, D v; (i) b, d, e V ; (ii) A B, E, D (ii) a b, e, d P, T, S, p, t, s, C B, E, D q, u, r. c b, e, d Q, U, R. (2)

! 1.18

!% #"

34

1.18. 3. 1.18 ") A % +* A I/ . 1.19 ) A, + ) P , A P @I/. 1 1.13 ) 2 A %% P %% A ,' P ,! ! ' A , P , ! '%8 8! ' %% %' ! $ # 1.12( ) ) A ⇐⇒ ) P E##+ 0) () ) 0 ) !&=> ) 0 3 :: &0)

J7@A ) ) #+ Æ

u x = 0 ! F5 &% 5F) A

A

A

A

A

A

3

i=1

i i

Aξ1 + Bξ2 + Cξ3 = 0.

(1)

/86< 35

(ξ , ξ , ξ )

[ξ , ξ , ξ ] ) (1) ) (1) [A, B, C] (A, B, C) ( §1.4

1

2

3

1

2

3

   A1 ξ1 + B1 ξ2 + C1 ξ3 = 0,

(2)

  A ξ + B ξ + C ξ = 0. 2 1 2 2 2 3

( (2) ( (2) (A, B, C). [A, B, C]. - &% . ! $ ) (1) (A, B, C) (1) [A, B, C] Au + Bu + Cu = 0. Ax + Bx + Cx = 0. (2) (0, 0, 1) (2) "& [0, 0, 1] u = 0. x = 0. (3) "& (A, B, 0) (3) [A, B, 0] Au + Bu = 0. Ax + Bx = 0. (4)∼(8) 1.5∼ 1.9. (4) ∼ (8) 1.5 ∼ 1.9 . 1)

1

2

3

1

2

3

3

3

1

1.

D+/

2

1

. /

!,)/8!,)

! 1.19

2

1.4

! 1.20

G48CDG;H?8& RP H (RP ) >@&>9>($IH8? RP H (RP ) > @:5AIAJEJK@:=B;8L 1)

2

2 ∗

2

2 ∗

!% #"

36

! 1.21 ! 1.22 2. 4+/ >.)/8>.-;+,/8!,) .! 1.23, /3-.)/*, ABC. .',, BC, CA, AB %*6($))/&+* P, Q, R, 958* B, C %*/+!) p, q +(<8* A B/!,+!) r +)

! 1.23 4+/ >.)/8>.) //,!) l , l +-4+76/* A , B , C . A , B , C . ,"+ B C . B C )0*( C A . C A )0*( A B . A B )0*/*$) ( Pappus +,). 4. 4+/ >.)/8>.) //*, ABC )/, BC, CA, AB )..%** α = 0, β = 0, γ = 0, M3%*( A, B, C )/!)),"+//!)$* ⇐⇒ ,..(-7* qβ − rγ = 0, rγ − pα = 0, pα − qβ = 0, , p, q, r *7C) 5. /K>. ($!&3/8>...'3(?4+) (1) +760*(, 6/*$)(3",!))5,7.) (2) -0,,)/*)7/%) (3) / A, B, C *#$)/*( D, E *761*(;? B, C, D $)5 C, A, E $)) 3.

1

1

2

2

2

1

1

1

2

1

2

1

1

2

2

2

1

2

2

1

§1.5 Desargues

+,

37

<,/* F , :, A, B, F $)5 D, E, F $)) (4) /<4/)0@*70) 6. 4+/ >.)/8>.) / OX, OY, OZ */,+!)( A, B *1+*(,.)6(* O, M* R * OZ +). *(5 RA, RB %*0 OX, OY * P, Q. ,"+ P Q 6( AB +)!&+*) 7. 4+/ >.)/8>.) / A, B *1+*( XY *+!))/ XY +-4* P, Q, / AP 0 BQ L, AQ 0 BP M . ,"+ LM 6( AB +)!&+*) F7 Desargues ?+ Desargues 1.13(Desargues ) , 1.13 (Desargues 0 ' ) ,! ' ! 35 0 0 Desargues 1.24, ABC A B C AA , BB , CC O. ! BC × B C = X, CA × C A = Y , AB × A B = Z. 0 X, Y, Z §1.5

Desargues

! 1.24

38

!% #"

A, B, C, A , B , C , O A 2 a, b, c, a , b , c ,

o.

A, A , O §1.2 1.8,

o = pa − p a .

o = qb − q b , o = rc − r c ,

p, p ; q, q ; r, r $52 

  qb − rc = q b − r c ,          

rc − pa = r c − p a ,

(1)

pa − qb = p a − q b .

! (1) 5 qb − rc q b − r c $B! BC $N! B C BC × B C = X. 5BN!& X Y rc − pa ! r c − p a Z pa − qb !

p a − q b .

(1) 5! $ §1.2 1.9, X, Y, Z 0@ Desargues &$ 1.24 $ O ABC A B C EF, ! X, Y, Z EFG. $ Desargues 0 ) () $ (&$ )

) Desargues Desargues . C 1.24, 10 10 % Desargues K@. Desargues ( =:) , 10 ) A Desargues OB C AZY , Desargues BXC. Desargues , ) Desargues , )

§1.5 Desargues

+,

HL1 01 1 ( 35

39

1.

AD,BE,CF. BC × EF = X, CA × F D = Y , AB × DE = Z. 0 X, Y, Z 1.25, Desargues & ABC DEF , 'Æ ABC #= G. ! X, Y, Z 1.14

1

ABC,

! 1.25

! 1.26

OX, OY, OZ A, B R OZ % RA, RB OX, OY P, Q. 0 P Q % AB 35 Desargues 1.26, OZ + R , 5 P , Q . ! P QR P Q R , $ O, #! A, B, C Æ P Q, P Q , AB C, $) P Q AB 1.15

1

1

1

1

1

C.

1

1

1

1

ABC A B C BC ×B C = L, CA ×C A = M , AB × A B = N ( 1.27). 0

(1) BC, B C , M N (2) ABC, A B C LM N 35 ABC A B C O. ! BCA B C A. BB ,CC , A A O, Desargues BC B C X, CA C A M BA B A N BC, B C , M N X. CA, C A , N L Y . AB, A B , M L Z. ABC A B C X, Y, Z ABC, A B C LM N

1.16

!% #"

40

(1), (2) &0

A1

1

! 1.27

P QRS( 1.28), ! ABC, + = BC × RQ, B = AC × RP, C = AB × P Q. 0 A , B , C 1.17

1

1

1

1

1

! 1.28

35

! P QR ABC, $ S, ) A , B , C $L( A , B , C ; D , A , E ; D , B , F ; E , F , C &( P QRS ! ABC 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

+, 41 8!8 A , B , C , D , E , F , 8"8' ' ABC 2. 1 G 1 "?&%"#> 1 1.18 a, b, P a, b a, b a × b, P c, c a × b. 2 (1) a, b P O, O l , l , l , l , l a, b A , A B , B ( 1.29). (2) P A l A , P B l B . (3) A A , B B Q. (4) P Q 35 (< .) §1.5 Desargues

1

1

1

1

1

1

1

1

2

1

1

2

3

3

2

3

2

1

1

3

2

3

2

3

3

! 1.29

. /

1.5

.! 1.30, /!) AB . CD 0 U , !) AC . BD 0 V , !) U V %*0 F, G, !) BF 0 AC L. "-+/!) LG,CF ,AU $*) 2. .! 1.31, 74#"+!,!) p -C#/ p +)761* A, B. #OC.9 AB, ,D!) AB . p )0* C. 3. .! 1.32, / a, b, c, d *#0,!)(#OC3 D + a, b )0*. c, d )0*(, D!!) l, :, l 6(;&0*) 4. .! 1.33, / ABCD 2!&0=(* X / BC +(!!)6( X %*0 AB, AC P, Q. 2!!)6( X %*0 DB,DC R, S. ,"+ P R . QS )0*/ AD +) 5. //&$)/*,2AM)53@$)AM ()"-+/,AM3$*) 1.

AD,BC

!% #"

42

! 1.30

! 1.31

! 1.32

! 1.33

.! 1.34, / A, B, C *#$)/*(P 2( C )!,+!)+).*(AP ×BC = X, AC × BP = Y . ,"+ XY 6(!&+*) 7. .! 1.35, /3/*, ABC, /,/, BC, CA, AB +%*4* A , B , C . / B C × BC = F , C A × CA = G, A B × AB = H. M F H × BB = M , F G × CC = N . "-+/ !) M G, N H, BC $*) 6.

! 1.34 8.

07 Desargues +,"-/*,)/, )$*)

! 1.35

+, 43 9. /-0,, EF GH )8*%*/2!&-0,, ABCD )-,+)07 Desargues +,"-+;&-0,,)0,/*)$*) 10. /'+(=&/*,)AM (2635**AM32635!)(.A8 Desargues +,?B,(3!>.. §1.5 Desargues

H A = D B ' =* "+6 ' MN 1. 2.1 P , P , P , P l(P ) % P = P , a, b, a + λ b, a + λ b. (P P , P P ) $ " MN, §2.1

1

1

2

3

2

4

1

1 2

2

3 4

(P1 P2 , P3 P4 ) =

λ1 . λ2

(2.1)

P , P , P , P $. 2.1 *! $ * 0- 2.1 l(P ) P 2 a + λ b(i = 1, 2, 3, 4, λ = λ ), a, b l(P ) A, B 1

2

3

4

i

(P1 P2 , P3 P4 ) =

35

i

(λ1 − λ3 )(λ2 − λ4 ) . (λ2 − λ3 )(λ1 − λ4 )

a+ λ b = a ,a + λ b = b .

1

a=

2

a λ2 − b λ1 , λ2 − λ1

P , P , P , P 1

2

3

a ,

!

b=

b − a . λ2 − λ1

4

b ,

λ2 − λ3 λ3 − λ1 a + b, λ2 − λ1 λ2 − λ1

a ,

b ,

a +

λ3 − λ1 b, λ2 − λ3 44

λ2 − λ4 λ4 − λ1 a + b, λ2 − λ1 λ2 − λ1 a +

λ4 − λ1 b. λ2 − λ4

1

2

(2.2)

0 C $ 2.1, §2.1

45

(P1 P2 , P3 P4 ) =

(λ1 − λ3 )(λ2 − λ4 ) . (λ2 − λ3 )(λ1 − λ4 )

6 $ 2.1, 3 %&& (>> ) * λ P $ 2.1 2 2.

i

(P1 P2 , P3 P4 ) =

i

P1 P3 · P2 P4 . P2 P3 · P1 P4

(2.3)

5N!& $ 2.1 D +* +*2D " 4! = 24 (P P , P P ) = r, 0 * D "? 2.2 D

(1) ! D (2) %! D$ r 1/r; % !B D$ r 1 − r. $ 2.2 2" 24 2 6 ( ( 4 6 ( DN EG 2.2. 1, 2, 3, 4 P , P , P , P . 1 2

3 4

                   

1

2

3

4

r = (12, 34) = (34, 12) = (21, 43) = (43, 21), 1 = (21, 34) = (34, 21) = (43, 12) = (12, 43), r 1 − r = (13, 24) = (24, 13) = (42, 31) = (31, 42),

1    = (31, 24) = (24, 31) = (42, 13) = (13, 42),   1−r    1    1 − = (14, 23) = (23, 14) = (32, 41) = (41, 32),   r     r   = (41, 23) = (23, 41) = (32, 14) = (14, 32). r−1

2.1 0- 3.

(2.4)

46

2.3

1% #"-6 " ( ⇐⇒ D 1, 0, ∞

2.2 30 6 D 3 (Æ 3 D 1, 0, ∞. 2.2 " (P P , P P ) = −1, P , P , P , P 2*" OPI, OPN. ( OP'. 2.2 P , P P , P ()OPC?, P , P P , P ()OP$%, ! P P , P , P OP. 4 2.1 " (P P , P P ) = −1, P , P , P , P 4 2.2 ( P , P , P , P 5(*"( ⇐⇒ " 6 D 3 ( ( D −1, 12 , 2. ! , r = −1, 1 2

3 4

1

2

3

4

1

3

4

1

2

3

4

4

2

1

2

3

1 2

3 4

1

1

1 = r = −1; r

2

3

2

3

4

4

1−r =1−

1 = 2; r

r 1 1 = = . r−1 1−r 2

) 6 D 3 ( ' (2.3) "% P = P , P P P P ? % PP PP = 1. " P P P PP PP = −1,

4 2.3 P , P , P % P "& P P P ⇐⇒ (P P , P P ) = −1. 4. " 1 2.1 P (3, 1, 1), P (7, 5, 1), Q (6, 4, 1), Q (9, 7, 1). (P P , 4

∞

1 ∞

3

2 ∞

1

1 ∞ 1 3

1 2

2

2 ∞

2 3

∞

1 2

1 2

∞

1

2

1

2

1 2

Q1 Q2 ).

2

30 P , P , Q , Q (O," "0). P , P * Q , Q . Q , ρ Q = P + λ P , Æ       1

1

2

1

2

2

6 7 3             ρ1  4  =  1  + λ1  5        1 1 1

1

2

1

1

1

ρ ),

(

1

1

1 2

0 C λ = 3, ρ = 4. 4Q Q Q Q , ρ Q = P + λ P , λ = −3, Æ Q §2.1

1

1

1

2

2

2

1

1

2 2

2

(P1 P2 , Q1 Q2 ) =

P

2.4

i

∈ l(P ) (i = 1, 2, 3, 4),

47

1

= P1 + 3P2 .

2

= P1 − 3P2 .

3 = −1. −3

(k = 0, 1, ∞)

(P1 P2 , P3 P4 ) = k

! 1 2.2 (P P , P P ) = 2 % P , P , P 2 (1, 1, 1), (1, −1, 1), (1, 0, 1). P 2 P = P + P , P = P + λP , 1

2

3 4

1

2

4

3

4

1

2

3

1

2

(P1 P2 , P3 P4 ) =

λ = 2, 1

λ = 2. P (3, −1, 3). 4 2.4 P , P , P l(P ) 3

0

1

∗

(P∗ P0 , P1 P ) −→ x

l(P ) R $ 2 2.4 P = P x = ∞; P = P x = 0; P = P x = 1. $ () 3 P , P , P "& 6 1 2.3 + P P P ! P P , P P , P P Q , Q , Q . !+ Q , Q , Q , ∗

0

1

∗

0

1 2 3

1

2

1

2 3

3 1

1 2

1

2

3

3

(P2 P3 , Q1 Q1 ) = k1 , (P3 P1 , Q2 Q2 ) = k2 , (P1 P2 , Q3 Q3 ) = k3 .

0 (1)

3

Q1 , Q2 , Q3

⇐⇒ k k k = 1; ⇐⇒ k k k = −1. 1 2 3

EE §1.2 1.6 ) P , P , P a, b, c. Q , Q , Q P P P !%#

Q : qb − rc; Q : rc − pa; Q : pa − qb (p, q, r)

P , Q # pqr = 0. P1 Q1 , P2 Q2 , P3 Q3

(2)

1 2 3

( )

1

2

3

1 2 3 1

i

j

2

3

1

2

3

1% #"-6 (P P , Q Q ) = k , " Q b + λ c, Q b + λ c, k = λλ , λ = − qr , λ = −k rq . Q Q : qb − k rc (k = 0, ,Q ≡ P ). 48

1

2 3

1 2

1

1

1

1

2

1

1

1

1

1

% k = 0, k = 0, Æ k k k = 0. ( ) 50 ( 2.1 2.2) 3

1

1

Q2 : rc − k2 pa,

2

1

1

2

Q3 : pa − k3 qb,

1 2 3

! 2.1 (1)

! 2.2

Q , Q , Q ⇐⇒ $ l, m, n, 1

2

3

l(qbi − k1 rci ) + m(rci − k2 pai ) + n(pai − k3 qbi ) = 0 (i = 1, 2, 3),

!

(n − k2 m)pai + (l − k3 n)qbi + (m − k1 l)rci = 0

$ P , P , P % pqr = 0, 1

2

(i = 1, 2, 3).

3

           

−

k2 m

l −k1 l

+

+

n

=

0,

−

k3 n

=

0,

=

0.

m

l, m, n, $Æ($ 0 1 −k1

−k2 0 1

−k3 = 0, 0 1

2

0 C

§2.1

49

Æ k1 k2 k3 = 1.

0 P Q , P Q , P Q ⇐⇒ k k k

(2)

1

1

2

2

3

3

1 2 3

= −1,

P1 (a1 , a2 , a3 ), Q1 (qb1 − k1 rc1 , qb2 − k1 rc2 , qb3 − k1 rc3 ),

§1.2 1.6, P Q

Æ

1

a1 a3 , , qb3 − k1 rc3 qb3 − k1 rc3 qb1 − k1 rc1 a1 a2 qb1 − k1 rc1 qb2 − k1 rc2

1

a2  qb2 − k1 rc2

a3

a3 a3 ,q c3 b 3

a a3 − k1 r 2 c2 b3

 a q 2 b2

  ,

a3 a1 a1 − k1 r , c3 c1 b1  a1 a2 a1 a2  . q − k1 r b1 b2 c1 c2

"% A , B , C a , b , c 5 |abc| 5 i

i

i

i

i

1)

i

P1 Q1 : [qC1 + k1 rB1 , qC2 + k1 rB2 , qC3 + k1 rB3 ].

Æ P Q 1

1

qCi + k1 rBi ,

i = 1, 2, 3.

P2 Q2 :

rAi + k2 pCi ,

P3 Q3 :

pBi + k3 qAi ,

i = 1, 2, 3.

P Q , P Q , P Q ⇐⇒ l, m, n(lmn = 0), 1

1

2

2

3

3

l(qCi + k1 rBi ) + m(rAi + k2 pCi ) + n(pBi + k3 qAi ) = 0,

i = 1, 2, 3

⇐⇒ (k3 qn + mr)Ai + (k1 rl + np)Bi + (k2 mp + lq)Ci = 0, i = 1, 2, 3.    P P : [A1 , A2 , A3 ],   2 3 P2 P3 , P3 P1 , P1 P2 ⇐⇒ |ABC| = 0 [B1 , B2 , B3 ], P3 P1 :     P P : [C1 , C2 , C3 ], 1 2

$ 1)

&FP@C?FQ9@AH@R?:FGH (@) GPGSC
1% #"-6

50

"( ⇐⇒

⇐⇒ (Ai , Bi , Ci )(i = 1, 2, 3)     

rm

k1 rl     ql

+k3 qn = 0, +pn = 0,

+k2 pm

= 0.

lmn = 0, #( ⇐⇒ 0 k1 r q

r

k3 q p = 0 ⇐⇒ pqr(k1 k2 k3 + 1) = 0 ⇐⇒ k1 k2 k3 = −1. 0

0@

0 k2 p

(1) " Q , Q , Q "& Æ Q , Q , Q "& 1

2

k1 =

3

1

P2 Q1 , P3 Q1

k2 =

P3 Q2 , P1 Q2

k3 =

2

3

P1 Q3 . P2 Q3

- (1) Menelaus ( P P P ! P P , P P , P P 2 Q , Q , Q , Q , Q , Q ⇐⇒ 1 2 3

1

2

3

1

2

2 3

3 1

1 2

3

P2 Q1 P3 Q2 P1 Q3 · · = 1. P3 Q1 P1 Q2 P2 Q3

- (2) " Q , Q , Q "& + Ceva ( P P P ! P P , P P , P P 2 Q , Q , Q , P Q , P Q , P Q ⇐⇒ 1

1 2 3

3

1

1

2

2

3

2

3

2 3

3 1

1 2

1

2

3

P2 Q1 P3 Q2 P1 Q3 · · = −1. P3 Q1 P1 Q2 P2 Q3

, MN (

p, [p]. ! =: ( S(p)

a, b, $ 1.8 , ( p p = a+λb 5 λ ∈ R 7. a, b ( S(p) * $ (

S(p)

0 C 51 1. ( 2.3 p , p , p , p ( S(p) % p = p . 2 a, b, a + λ b, a + λ b, (p p , p p ) $ " MN, λ §2.1

1

2

1

3

4

2

1

1 2

2

3 4

(p1 p2 , p3 p4 ) =

1

λ2

.

(2.5)

( S(p) p 2 a+λ b(i = 1, 2, 3, 4, λ a, b S(p) 2.5

i

(p1 p2 , p3 p4 ) =

i

1

= λ2 ),

(λ1 − λ3 )(λ2 − λ4 ) . (λ2 − λ3 )(λ1 − λ4 )

(2.6)

2.6 ( S(p) p s $ P 2.

i

i

(i = 1, 2, 3, 4),

(p1 p2 , p3 p4 ) = (P1 P2 , P3 P4 ).

35

(2.7)

p , p , p , p a, b, a + λ b, a + λ b, s c. P 2 1

 a2 a3  P1 c2 c3

i

4

3

a3 a1 , c3 c1

0@

2

4

a1 a2 , c1 c2

1

  b b  , P2  2 3 c2 c3

b3 b1 , c3 c1

P3 (P1 + λ1 P2 ),

P4 (P1 + λ2 P2 ).

(p1 p2 , p3 p4 ) =

λ1 = (P1 P2 , P3 P4 ). λ2

2

b1 b2 , c1 c2

  ,

P l(P ) P l S 2 p (i = 1, 2, 3, 4). 2.5

i

i

i

(p1 p2 , p3 p4 ) = (P1 P2 , P3 P4 ).

$- :6 2.7 1 "1 0% DH( 24 *D 3. (1) %( 1'

52

1% #"-6 ( S(p)

S 3( (x , y ). x = x , y = y p D1' −k * (y − y ) + (−k)(x − x ) = 0, Æ y + (−k)x + b = 0 5 k ∈ R. 2.8 %( k 1' p (i = 1, 2, 3, 4), 0

0

0

0

0

i

0

i

(p1 p2 , p3 p4 ) =

(k1 − k3 )(k2 − k4 ) . (k2 − k3 )(k1 − k4 )

(2.8)

%( (@ %( S(p) p x )6( α tanα . ( (2.8) 5 (2)

i

i

(i = 1, 2, 3, 4).

k

i

=

i

(p1 p2 , p3 p4 ) = = =

(tanα1 − tanα3 )(tanα2 − tanα4 ) (tanα2 − tanα3 )(tanα1 − tanα4 ) sin(α3 − α1 ) sin(α4 − α2 ) sin(α3 − α2 ) sin(α4 − α1 ) sin(p1 p3 ) sin(p2 p4 ) , sin(p2 p3 ) sin(p1 p4 )

(p p ) p p ( 2.9 p , p , p , p %( S(p) i j

i

1

4

2

j

3

4

(p1 p2 , p3 p4 ) =

sin(p1 p3 ) sin(p2 p4 ) . sin(p2 p3 ) sin(p1 p4 )

(2.9)

%( p (i = 1, 2, 3, 4), p ⊥p . p , p p , p ( ⇐⇒ (p p , p p ) = −1. 4. 2" 30 , *+ Æ 2.3 $$ (2 7$ 6), 1 2.4 A, B, C, D * P * 0 P (AB, CD) = 35 2.3, P, P P A* ((@#- 2.10 p ∈ S(p) (i = 1, 2, 3, 4), 1

2.6

i

2

1 2

3

3 4

1)

i

(p1 p2 , p3 p4 ) = k

1)

(k = 0, 1, ∞)

&QIHHI?JBPJB P (AB, CD) GS (P A, P B; P C, P D).

4

3

4

0 C ! §2.1

1 2

53

! 2.3

l , l , l 2 a = [3, 1, 1], b = [7, 5, 1], d = [9, 7, 1], % (l l , l l ) = −1. l l d l , l * d = a − 3b. l c = a + λb. 1 2

2.5

1

3 4

2

4

3

4

1

2

(l1 l2 , l3 l4 ) =

l

3

λ = −1 ⇐⇒ λ = 3. −3

3

c = [3, 1, 1] + 3[7, 5, 1] = [24, 16, 4] ≡ [6, 4, 1]. l3

4

6x1 + 4x2 + x3 = 0.

p , p , p ( S(p)

2.7

0

1

∗

(p∗ p0 , p1 p) −→ x

S(p) R $ p = p x = ∞, p = p x = 0, p = p x = 1. $ ( $ 5. % ∗

0

. /

1.

1

2.1

/ P (2, 1, −1), P (1, −1, 1), P (1, 0, 0), P (1, 5, −5). "-+ P , P , P , P 0*$) 1

-, (P P , P P ). 1

2

3

4

2

3

4

1

2

3

4

1% #"-6 2. / P , P %*2 x 3( y 3+)635*( P 28-* 1 ).5+)635*(5 (P P , P P ) = r. , P ) - 1) 3. / 1, 2, 3, 4, 5, 6 2 6 &#/)335$)*)"-+ 54

1

1

2

3

2

3

4

4

(1)

(12, 34)(12, 45)(12, 53) = 1;

(2)

(12, 34)(12, 56) = (12, 36)(12, 54);

.' (12, 34) = (14, 32), < (13, 24) = −1. / A, B, P, Q, R 276)335$)*(95 (P A, QB) = (QR, AB) = −1. "-+ (P R, AB) = −2. 5. /!) l , l , l , l ).../("-5,$*-,0C (l l , l l ). (3)

4.

1

2

3

4

1 2

3 4

(1)

2x − y + 1 = 0, 3x + y − 2 = 0, 7x − y = 0, 5x − 1 = 0;

(2)

x − y = 0, 2x + y = 0, x + y = 0, 3x − y = 0;

(3)

x1 − x2 = 0, 3x1 − x2 − 2x3 = 0, x1 − 3x2 + 2x3 = 0, x1 − x3 = 0.

6.

/!) l , l , l )..%** 1

3

4

2x1 + x2 − x3 = 0,

x1 − x2 + x3 = 0,

x1 = 0,

95 (l l , l l ) = − 23 , , l )..) 7. "-+!) a x + 2h xy + b y = 0 I.%;!) a x + 2h xy + b y = 0 )0 9,22 a b + (λa b −−λ2h)(λh −=λ0.) 78+75 (λ − λ )(λ − λ ) = −1 . 2(λ λ + λ λ ) = (λ + λ )(λ + λ ) 0K-. 7DB+,) 8. ,!) ax + 2hxy + by = 0 41*)#E%)..) 9. "-;CJ+,<+.! 2.4, (!0)K AB ) * O - D ,F1K CE, DF , . EF, CD 0 AB G, H, < GO = OH. 1 2

3 4

2

1

1 2

2

2 1

1

1

2

2

2

2

2

2

1 2

1

3

2

4

2

3

1

4

2

1 2

3 4

1

2

3

4

2

2.4 G@TKLQG@TML2LDN 3% Æ &% ( +7 §2.2

8<0*,.8<0),)I.O 55 DF + DF +OP' 2.11 2.11 ! !! ' '' ! '

) 0 Æ E 0 2.11, -0 2.11 . 35 2.11 2.5, ABCD, 0 (ss , tt ) = −1. §2.1, 2.6, §2.2

X(ss , tt ) = (AB, P Z),

0 (AB, P Z) = −1. A, B, C, D " §1.2, 1.4, 6# a, b, c, d a + b + c + d = 0,

Æ a + b = −(c + d).

a + b c + d AB CD Æ Z = AB × CD. Z a + b ! c + d. X a + d ! b + c; Y a + c ! b + d. a − b = (a + d) − (b + d),

5BN! AB XY ) a − b AB XY Æ P .

! 2.5

! 2.6

56

$ A, B, P, Z a, b, a − b, a + b,

1% #"-6

(AB, P Z) = −1 = (ss , tt ).

0 Y, Z 0@ 35 2.11 2.6, ' x, 0 (SS , T T ) = −1, §2.1 2 2.5, 0 (ab, pz) = −1.

(ab, pz) = (y × a, y × b; y × p, y × z) = (cd, qz),

(cd, qz) = (SS , T T ) = (ba, pz),

(ab, pz) = (ba, pz) =

1 , (ab, pz)

Æ [(ab, pz)] = 1. (ab, pz) = 1, (ab, pz) = −1. 0' Æ 0@

-$ 2.5 2.6 Æ &%( §2.1 2.7, # 2 4 2.8 ! 4 2.8 % !( ' ' !+! ( '+' !(! '(' $ 2 2.5 2.6 Æ &%( 4 2.9 4 2.9 % !( ' '( + ! !+ ' !!! ''' $ 2 2.5 2.6 6 Æ &%( 2ÆÆ "DE

OP'HL1 1. &% 1 1 2.6 2.7, l P , P , P , P . 2

1

2

3

4

8<0*,.8<0),)I.O 2 (1) P , P l , l , l (2) P l l , l B, D.

§2.2

1

3

3

2

1

1

2

57

×l2 = A.

1

2

! 2.7 P D, P B C, AC l P . P Æ 0 $ ABCD P , P ! P , P $! P P 2 2.8, (P P , P P ) = −1.

2 2.9 )! (P P , P P ) = −1 ⇐⇒ P , P ! P , P +!$! (- 2. 0 1 2.7 T P, T Q O * P R O QN ⊥RP . 0 QN T R 35 T R × QN = X, 0 X QN (1) $QR, QP QN QT ( T Q × P R = S, Q(SN, RP ) = −1, (SN, RP ) = −1, T (SN, RP ) = −1. (3)

1

2

4

4

1

1 2

1 2

1 2

3

2

4

1

2

3 4

3 4

4

! 2.8

3

! 2.9

1% #"-6

58

(2)

QN $ T (S, N, R, P ), QN × T P = Y , ∞

(QN, XY∞ ) = T (SN, RP ) = −1.

X QN 1 2.8 0 HFG?( & 35 2.9, ABCD H E F AB CD F ( EF AD, BC P, Q. 0 P, Q AD BC

! EAF D, AD×BC = G . 2 2.8, (BC, QG ) = −1, Q BC 2 2.9, (AD, P G ) = −1, P AD 1 2.9 X ABC AD BX, CX ! Y, Z. 0 AD ∠Y DZ. 35 DY CX P . ! CDXY , A, B, P ! 2 2.9, (CX, P Z) = −1, Æ D(CX, P Z) = −1. DX⊥DC, # DX(Æ AD), DC DP , DZ ( ∞

∞

∞

! 2.10

. /

1. 2.11).

2.2

/ XY Z 28<0*, ABCD )/,/*,( XZ %*0 AC, BD L, M (.! #7 Desargues +,"-+ Y Z, BL, CM $*)

! 2.11

! 2.12

!<E8,)#"/. 59 2. / ABCD *0,,( AB × CD = E, AD × BC = F, AC × BD = O. ( O D l//AB 0 CD, EF G, H(.! 2.12). ,"+ OG = GH. 3. / A, B, C 28<0),)/&$))8*(*8 A, B . M, C I.$ E ),"+6( A, B )/8))0*/ M . C )/8*).)+) 4. 4+ 3 .)/8>.) 5. / XY Z 28<0*, ABCD )/,/*,( P 2+).*)!) l "(* X, !)8 l, XP .( X ),I.$E/.D=+<%*"(* Y, Z )!) m, n. "-+/! ) l, m, n $*) 6. /3/*, P P P , Q , Q , Q *$)/*%*I/,+)M (R Q , P P ) = −1, (R Q , P P ) = −1, (R Q , P P ) = −1. ,"+ P R , P R , P R $* (.! 2.13). §2.3

1

2

2

3

1

2

3

3

1

3

1

2

3

1

2

1

1

2

2

3

1

2

3

3

! 2.13

/E+)M AB ) * C -C#/!) AB +)!* M . 977!C,D( M 5 - AB )!)) 7.

HJFIL29:@I + 5 EF 2.4 EF. (1) (' ,&% 2.14, §2.3

s(A, B, C, · · · )∧S(a, b, c, · · · ).

' , 2.15, (2)

(S)

s(A, B, C, · · · ) ∧ s (A , B , C , · · · ).

1% #"-6

60

S EF. (3) (' , 2.16, (s)

s EFG.

S(a, b, c, · · · ) ∧ S (a , b , c , · · · ).

! 2.14

! 2.15

! 2.16

$ 2.4 $%:6&% "DN $ )

1. Poncelet 2.5(Poncelet) [π], [π ] " n [π ] (i = 1, 2, · · · , n),

i

[π] ∧ [π1 ] ∧ · · · ∧ [πn ] ∧ [π ],

$' [π] [π ] , [π] ∧ [π ]. $ 2.5, Æ+ "C 4 2.10 (1) (2) " (3) $%:6&% 2.5 $ L+ - 2. Steiner 2.6(Steiner) ϕ : [π] → [π ] $" ϕ &% ϕ [π] [π ] ,

!<E8,)#"/.

§2.3

35

61

[π] ∧ [π ].

⇐⇒ Steiner =⇒ ⇐= ϕ : [π] → [π ] Steiner 0 ϕ 2.17, P , P , P l(P ) P l(P ) % ϕ(P ) = P (i = 0, 1, 2), ϕ(P ) = P . 2.12 Poncelet

0

1

2

i

i

! 2.17 (P0 P1 , P2 P ) = (P0 P1 , P2 P ).

- P P , P P ; P P , P P , P P ×P P = Q , P P ×P P = Q . Q Q P P m Q, P Q l P , P P m Q . 0

0

1

0 2

0 1

0 2

0 1

0

0 1

(P0 )

0 2

1

0 0

0 2

2

1

2

= m,

0

(P0 )

l(P0 , P1 , P2 , P ) ∧ m(Q0 , Q1 , Q2 , Q) ∧ l (P0 , P1 , P2 , P ),

Poncelet l(P0 P1 , P2 P ) ∧ l (P0 P1 , P2 P ),

(P0 P1 , P2 P ) = (P0 P1 , P2 P ) = (P0 P1 , P2 P ). P = P .

Æ P ∈ l(P ), & P = ϕ(P ) % Æ ϕ Poncelet 0(( $ (G-0@

1% #"-6 2.12 0 Steiner l(P )∧l (P ), &% P P , l(P ) &% P , l (P ) &% P . $ G, (!(&% 2.12 0 )

2.13 &%! 3. &

" ( ) (" ( )). 01 P 2.14 ⇐⇒ &%

35 + 2.14 20(

(1) 0A0 ( (2) 0A0 ( ( 1 2.10(Pappus ) l

A , B , C (i = 1, 2). 62

i

i

i

B1 C2 × B2 C1 = L, C1 A2 × C2 A1 = M, A1 B2 × A2 B1 = N.

0 L, M, N 35 2.18,

(A1 )

(C1 )

(B1 , D, N, A2 ) ∧ (O, C2 , B2 , A2 ) ∧ (B1 , C2 , L, E), (B1 , D, N, A2 )∧(B1 , C2 , L, E).

i

i

i

!<E8,)#"/. & B $ 2.14, §2.3

63

1

(B1 , D, N, A2 )∧(B1 , C2 , L, E),

DC , N L, A E Æ L, M, N 2

2

! 2.18

J7

! 5 6(!( 2.7 & % (x , x ), (x , x )

0QJ' 1

   ρx1 = a11 x1 + a12 x2 ,   ρx = a x + a x , 21 1 22 2 2

a12 = 0, a22

a11 a21

2

1

2

ρ = 0

(2.10)

. 2.15 Steiner ⇐⇒ 35 =⇒ l(P ) l (P ) $ P → P (i = 1, 2, 3) ! P → P Steiner

i

i

(P1 P2 , P3 P ) = (P1 P2 , P3 P ).

l, l P , P , P , P 2

1

2

3

(a1 , a2 ), (b1 , b2 ), (a1 + λ0 b1 , a2 + λ0 b2 ), (a1 + λb1 , a2 + λb2 ),

% P , P , P , P 1

2

3

(a1 , a2 ), (b1 , b2 ), (a1 + λ0 b1 , a2 + λ0 b2 ), (a1 + λ b1 , a2 + λ b2 ).

1% #"-6

64

λ λ0 = 0 . λ λ

(∗)

, (x , x ), (x , x ) P, P 1

2

1

2

x1 x2 = , a1 + λb1 a2 + λb2

λ=

a2 x1 − a1 x2 . −b2 x1 + b1 x2

λ =

a2 x1 − a1 x2 . −b2 x1 + b1 x2

λ, λ ( (*),

λ0 (−b2 x1 + b1 x2 ) λ (−b x + b x ) = 0 2 1 1 2 , a2 x1 − a1 x2 a2 x1 − a1 x2

   λ0 (−b2 x1 + b1 x2 ) = ρλ0 (−b2 x1 + b1 x2 ),   a x − a x = ρ(a x − a x ), 2 1 1 2 2 1 1 2

ρ $5 x , x 1

2

   ρx1 = a11 x1 + a12 x2 ,   ρx = a x + a x , 21 1 22 2 2

ρ = 0, a &% 5 ij

a12 = 0. a22

a11 D = a21

, l & l F Steiner $#4 ⇐= / (2.10). l ! P (x , x ), $ (2.10) !$ (ρx , ρx ). P l !

1

1

2

2

!<E8,)#"/. $ D = 0, $ (2.10) 05

§2.3

65

   σx1 = A11 x1 + A21 x2 ,

(2.11)

  σx = A x + A x , 2 12 1 22 2

σ = Dρ , A a D 5 l l ) (2.10) 5! l(P ) l (P ) $ 0 (2.10) 5:6 P , P , P , P l(P ) % P = P . (2.10) l (P ) P , P , P , P , P = P . ( 2 ij

ij

1

1

1

2

2

3

1

2

P1 (a1 , a2 ), P2 (b1 , b2 ), P3 (a1 + λ0 b1 , a2 + λ0 b2 ), P (a1 + λb1 , a2 + λb2 ), P1 (a1 , a2 ), P2 (b1 , b2 ), P3 (a1 + λ0 b1 , a2 + λ0 b2 ), P (a1 + λ b1 , a2 + λ b2 ).

! ρ1 a1 = a11 a1 + a12 a2 , ρ2 b1 = a11 b1 + a12 b2 ,

ρ3 (a1 + λ0 b1 ) = a11 (a1 + λ0 b1 ) + a12 (a2 + λ0 b2 ).

5(5I (ρ3 − ρ1 )a1 = (λ0 ρ2 − λ0 ρ3 )b1 ,

$ $

(ρ3 − ρ1 )a2 = (λ0 ρ2 − λ0 ρ3 )b2 .

$ P P 1

2

a1 a2

ρ3 − ρ1 = 0,

b1 = 0, b2 λ0 ρ2 − λ0 ρ3 = 0.

λ0 ρ1 . = λ0 ρ2

2

3

1% #"-6

66

! λ ρ1 . = λ ρ2

(P1 P2 , P3 P ) =

λ λ0 = 0 = (P1 P2 , P3 P ). λ λ

-/ (2.10) Steiner 0@ -"& &% (2.10) 55 x =

!

a11 x + a12 , a21 x + a22

|aij | = 0,

axx + bx + cx + d = 0,

(2.12)

1 5 "& "& 2 "2 2.11

l

2, 4

−1, 1

l

(2, 1)

←→

(−1, 1)

(4, 1)

←→

(1, 1)

(1, 0)

←→

(1, 0)

   ρx1 = a11 x1 + a12 x2 ,

(2.10)

  ρx = a x + a x , 21 1 22 2 2

a : a : a : a Æ 2( (2.10), 11

12

21

22

(2, 1) ←→ (−1, 1)    −ρ1 = 2a11 + a12   ρ = 2a + a 1 21 22

(x1 , x2 )

(x1 , x2 )

5

(2.12)

ad − bc = 0.

(4, 1) ←→ (1, 1)   ρ2 = 4a11 + a12  ρ = 4a + a 2

21

22

(1, 0) ←→ (1, 0)   ρ3 = a11  0=a 21

!<E8,)#"/. a = 0, ( 5 §2.3

67

21

a11 : a12 : a21 : a22 = 1

5 "Æ x = x − 3.

J − 3J0J1.

   ρx1

=

x1 − 3x2 ,

  ρx 2

=

x2 ,

. /

2.3

,#"/.B(:!) l +-1* 0, 1, 2 )/*U%/.!) l +-1* −1, 0, −2 )/*) 2. / 1 .(,D,!)+635*)/.*) 3. ,#"/.B(:!) l + - 1* 1, 2, 3 )/*U%/.!) l +)/*(, - 1% **

1.

(1)

4,3,2;

74*

(2)

−1, −2, −3;

(3)

1,2,3.

. l (P ) +)///.* A(3, 1), B(0, 5), C −1, − 13 ←→ A (2, −3), B (6, −7), C (1, 4) R/2G(;!A+ l ( l )!&#"/..*3!. 5. 9/* )B<"-+, 2.14. 6. .'/*, ABC ), BC, CA, AB %*6(//!!)+)/* P, Q, R, M8* B, C -/!,+!)+),"+8* A B/!,+!)+) 7. /3!&-.)/*, ABC )/, BC, CA, AB %*6(/&+* X, Y, Z, 58* B, C %*/,+!) m, l + (! 2.19). "-+)4 Y (A)∧ )4 Z(A). 4.

l(P )

! 2.19 ! 2.20 0* O, +* S , S . O $)(.!) l 6(2!+* M , %*

/+!) l , l 0 l , l * A , A (! 2.20). 8.

1

1

2

1

2

2

1

2

68

"-+!) S A , S A 1

1

2

2

1% #"-6 )0*HI*!,!))K3+/!)/3!,2/"(* O?

HJ9:KJ §2.3 +7 - 1. # &%& % 2.8 . - . !Æ& l(P ) ∧l(P ) !(( S(p)∧S(p ). )CE&%&% $ &%$.5 &% &% 7-,6 2. * 5 (2.10), +Æ* 2.9 axx + bx + cx + d = 0 (ad − bc = 0) x, x §2.4

K'34 5$ &%*

.

2.16

⇐⇒

$

35

aλλ + bλ + cλ + d = 0

(ad − bc = 0).

λ, λ

(2.13)

(*$-" 0 " ( =⇒ ! (P )∧(P ). &% A, B, P ∈ (P ), P = A + λB, λ ∈ R. $ A + λ B ←→ A + λ B (i = 1, 2, 3, λ , λ )

i

i

i

i

!<#"-6 ' A + λB A + λ B &% Steiner §2.4

69

(λ − λ3 )(λ2 − λ ) (λ1 − λ3 )(λ2 − λ) = 1 . (λ2 − λ3 )(λ1 − λ) (λ2 − λ3 )(λ1 − λ )

$2 λ λ (2.13). ad − bc = 0, ,$ ad − bc = 0 2 ab = dc = t, b = at, d = ct ( (2.13), λ ∈ R, λ = −t, $#4 ⇐= $ (2.13) 5 30 Steiner Æ (2.13) ' $%:6&%0>

(2.13) 798. * 5 (2.10) $ (2.13) %() ( 2.16 0 "DE!&-2# (2.13) 2Æ $6

- $, 1. &% !, :6&%&% 2.17 3LJ&%2 $ &% 35 (2.13) % λ = λ ,

aλ2 + (b + c)λ + d = 0

(ad − bc = 0).

(2.14)

&*3L J &% 2.

! (2.14), (1) λ , λ , (2.13) & % (2) λ , &%!) &% (3) Æ : λ , λ , Æ :&% 1

1

1

2

2

1% #"-6 2.10 ϕ , ϕ &% K) ; , ϕ &% RL) ; , ϕ Æ:&% MN) . 3. &% $$(*$A$- 2.18 "$ &% &%&% ÆO+. 35 X, Y &% P, P &% (&%). 0 (P P , XY ) = X, Y, P, P 2 x, y, x+y, x+µy, * 0, ∞, 1, µ. ( (2.13) 5 µ = − cb , # c = . (P P , XY ) = − b (2.13) * −c/b. 2.19 A$&%* α, λ, λ &%* 1 1 − = k(). λ−α λ −α 35 S05 70

λλ − (α −

α Æ ($ 5 %

1 1 )λ − (α + )λ + α2 = 0, k k

aλ2 + (b + c)λ + d = 0 λ2 +

α2 =

d , a

d b+c λ+ = 0 a a 2α = −

b+c a

c b d λλ + λ + λ + = 0, a a a c c λλ − (2α + )λ + λ + α2 = 0. a a k=−

a (= c + aα

),

(∗)

!<#"-6 71 "Æ (∗) 5 1 2.12 A, B, C (P )∧(P ) A, B, C 2 B, C, A. 0 (*$ 35 A, B, C 2 α, α+γ, γ, " A, C * * λ §2.4

A:

α + 0γ

=⇒ λ = 0,

B : α+γ

=⇒ λ = 1,

C:

=⇒ λ = ∞.

0α + γ

λ

←→

λ

A

0

1

B

1

∞ C

C

∞

0

(P )∧(P ) 5

B

A

aλλ + bλ + cλ + d = 0 ⇒ λ = −

bλ + d . aλ + c

*D2( (∗) J

d =⇒ c : d = 1 − 1, c a+c 0=− =⇒ a : c = 1 − 1, b+d b 0 = − =⇒ b = 0, a

A ←→ B :

1=−

B ←→ C : C ←→ A :

J

J J − 1J1.

a:b:c:d=1 0

5$

λλ − λ + 1 = 0.

:&%(*$

. /

1.

/- 0, 2, −2 *FC)*8%/.-

2.4

1 4 , , −2 2 3

*FC)*(,#"-6B)

(∗)

1% #"-6

72 2.

,/ !<#"-6)#->VFC)

(1)

λλ − 6λ + λ + 6 = 0;

(2)

λλ − 2λ + 1 = 0;

(3)

2λ + λ + 1 = 0.

3. / A , A , A * - 1/*,( O(1, 1, 1), A O 0 A A A, P 2 A A +).*( 0 A A Q, QA 0 A A P . .' P, P )'%-1%** (0, λ, 1), (0, λ , 1). ,# "-6 (P )∧(P ) )...#->V) 4. / A, B, C 276)$)*53 1

PO

1

2

3

2

2

2

3

1

3

2

3

(A, B, C, P, Q, R)∧(B, C, A, Q, R, X).

,"+ X . P K7) 5. /!<#"-6)&76#->V)FC%** 1, −1, -!//.>V.&#>V)0C* 2, ,#"-6B) 6. .! 2.21, / P, P ; Q, Q 2* l(P ) +#"-6)//.*( E 2#-*( V, V 2( E )!)+)-5*( P V × P V = P , QV × Q V = Q . ,"+ P Q . l )0 *22!&#-*)

! 2.21 ! 2.22 7. /* )#"-62BQ?)( E *#-*( P −→ P 2!//.*)-L P =L !* )*1(/;&#"-6/3 P −→ R(.! 2.22). ,"+ (EP , P R) = −1. 8. .! 2.23, / A, B, C, D; A , B , C *!) l +76)*)/#"-6

/(,D* D )/.*)

(A, B, C, · · · )∧(A , B , C , · · · )

! 2.23 HJFIL2@K §2.5

!<E8,)/7 73 $ §2.4 $$(*$&% &%* * −1 Æ S 1. ( @ f : R −→ R (R), ∀x ∈ R, f : x −→ −x (Æ% y = f (x) = −x). @ (1) f +D+ R, %6DÆ f R R $ (2) f % f = f. ∀x ∈ R, f (x) = f (x), Æ f f ( f = I, I 2#Æ + f -,2). §2.5

−1

−1

−1

−1

2

! 2.24 , $ f (∞) = ∞ R R, f & ! l(P ) l(P ) f (2) #

(1) X l(P ) &% f (X) = X ∈ l(P ) X (2) X l(P ) &% f (X) = X ∈ l(P ) 2 X

) X, " l(P ) &%

l(P ) &%&% 2.11 ! "&% " "! &%&% 2 S.

−1

1% #"-6 2.11 f [π] 2" x ∈ [π], f (x) = f (x), f [π] S. 2.11, 72$ * 2. 2.20 ! A + λB A + λ B ⇐⇒ f &%* λ, λ 74

−1

aλλ + b(λ + λ ) + d = 0,

ad − b2 = 0.

(2.15)

0> (2.15) 5 5 λ, λ F *

TSPU 1. & $ (2.15) 5 2.21 $ &% ! 4 2.11 &% P , P (i = 1, 2, 3) " ⇐⇒ *

i

35

p1 p1 p2 p2 p3 p3

Pi , Pi

p1 + p1 p2 + p2 p3 + p3

i

1 1 = 0. 1

(2.16)

" ⇐⇒ * (2.15), Æ api pi + b(pi + pi ) + d = 0

(i = 1, 2, 3)

5"( a, b, d $ ⇐⇒ (2.16) 5 4 2.12 " &%* p , p (i = 1, 2), $! ⇐⇒ i

i

λλ p p 1 1 p2 p2

λ + λ p1 + p1 p2 + p2

1 1 = 0. 1

(2.17)

!<E8,)/7

§2.5

75

1 &%* ' 2 &%* $ 2.13

1

λ, λ ,

λλ 1 1× 2 0×2

"

λ + λ 1 1+ 2 0+2

1 2

0

2.

$

(2.17),

1 1 = 0, 1

λλ + λ + λ − 2 = 0.

& f 2.

(a, b, c, · · · )∧(a , b , c , · · · ).

, f

(a, a , b, c, · · · )∧(a , a, b , c , · · · ),

$) a, a &% f & a, a &% f QV. 2.22 "2&% 35 f : (P )∧(P ) P ←→ P (i = 1, 2, 3) f &% f $!Æ

i

i

(P1 , P2 , P3 , · · · )∧(P1 , P2 , P3 , · · · ).

P , P 1

1

(P1 , P1 , P2 , P3 , · · · )∧(P1 , P1 , P2 , P3 , · · · ),

(P1 P1 , P2 P3 ) = (P1 P1 , P2 P3 ).

(2.18)

P (P ) &%&% (P ) &% Q. 0 Q = P . 2

2

(P1 , P1 , P2 , P2 , P3 , · · · )∧(P1 , P1 , P2 , Q, P3 , · · · ),

1% #"-6

76

(P1 P1 , P2 P2 ) = (P1 P1 , P2 Q) = (P1 P1 , QP2 ),

Q = P , 2

(P1 , P1 , P2 , P2 , P3 , · · · )∧(P1 , P1 , P2 , P2 , P3 , · · · ).

Æ P , P 0 f &% f 0@ 4 2.13 &% P , P (i = 1, 2, 3) " ⇐⇒ 2

2

i

(P1 P1 , P2 P3 ) = (P1 P1 , P2 P3 ). (2.19)

i

(2.19)

5!&

S !WX'(

(2.15) % λ = λ !&%*

aλ2 + 2bλ + d = 0

(ad − b2 = 0),

(2.20)

#- 2.23 &% 2.23, 5&% K)S MN )S A$ 2.24 ⇐⇒ & %% &%&% 35 =⇒ f : (P ) → (P ) E, F &%P, P (P = P ) &%$ §2.4 2.18 (P P , EF ) = k().

P, P ; E, E; F, F &%$&

(P P , EF ) = (P P, EF ),

Æk

k = 1. # k = −1. ⇐= "&% P, P , & (P P , EF ) = −1,

2

= 1,

(P P , EF ) = (P P, EF ).

!<E8,)/7 77 & P, P ; E, E; F, F "$ P 0@ 2.24 )* −1. 1 2.14 A, A ; B, B &% E, F &%0 A, B; A B ; E, F "+ &% 35 0 &$ §2.5

(AB , EF ) = (A B, EF ),

(AB , EF ) = (BA , F E) =⇒ (EF, AB ) = (F E, BA ),

$&- 1 2.15 &%* 1, 2. 2 aλλ + b(λ + λ ) + d = 0. &% aλ + 2bλ + d = 0. M

 d,   1·2= a   1 + 2 = − 2b , a

=⇒

a : b : d = 2J − 3J4. #

2

   d a = 2,   −2 b = 3, a

2λλ − 3(λ + λ ) + 4 = 0.

S '$ !" 35 $ Desargues

2.25(Desargues 2.25,

)

! P, P ; Q, Q ; R, R .

A, B, C, D

s

(B)

(A)

(P, P , Q, R) ∧ (X, P , D, C) ∧ (P, P , R , Q ),

(P, P , Q, R)∧(P, P , R , Q ).

(P P , QR) = (P P , R Q ) = (P P, Q R ),

P, P ; Q, Q ; R, R "

ABCD

1% #"-6

78

! 2.25

1

! 2.26

P, P ; Q, Q l(P ) R l(P ) R R . 2 (1) l X, XP, XP ; (2) Q XP, XP A, D; (3) AR XP C, Q C XP B (4) BD l R 0 $ ABCD l ' P, P ; Q, Q ; R, R l ABCD $! 2.25, P, P ; Q, Q ; R, R "

2.16

. /

2.5

74!<E8,+/7)//.>V)FC8%* 2, 2; 1, 4. ,/7..) , 1 . /7)#->V)FC(-&+5,2$!9/7) 3. 74!<E8,+)/7)&#->VFC* 1. 2.

§2.6 (1)

1<#"-6

79

2, 3,

.. αt + 2βt + γ = 0(β − αγ = 0) )A( ,/7..) 4. 4++, 2.25 )/8>.) 5. / A, B, C, D 2$)*5 (AB, DP ) = (AB, P C). ,"+ P 39(?IM5. A, B I.$E) 6. / A, B, C, D, P *76)$)Æ*(5 (AB, DP ) = (AB, P C). "-+ P 2R A ↔ B, C ↔ D 4A+)/7/)#-*) 7. / (A, B, C, D, E, F )∧(B, C, D, A, E, F ). "-+ E, F *R A ↔ C, B ↔ D 4A+) /7 )#->V) 8. ,"+)4)/7=3!//.!)7/$!).?G3HO//.!)7/$!. 2

(2)

2

PJ9:KJ + (") 6 &% - 2.12 "$" EF. 2.13(Steiner) π, π "" ϕ : π → π (1) ϕ $ (2) ϕ (3) ϕ :6 ϕ " π " π -. 2.14 " π π

$5'  §2.6

  ρx1 = a11 x1 + a12 x2 + a13 x3 ,    ρx2 = a21 x1 + a22 x2 + a23 x3 ,      ρx = a x + a x + a x , 31 1 32 2 33 3 3

|A| = |aij | = 0, ρ = 0.

(2.21)

" π π 0QJ'. (x , x , x ), (x , x , x ) A / 1 , π = π , % (2.21) " π 0

1

2

1

3

QJ'

.

2

3

1% #"-6 / "$# - 2.26 π π

f : π −→ π / 5 (2.21). π A, B f π A , B , A, B a, b f A , B ρ a , ρ b . f (a + λb) = ρ a + λρ b . 2.26 )/ %%:6 ** 2.27 "/ 2.28(M¨obius) π, π

P , P , P , P P , P ,P , P π π ( ( (&" ). / f : π −→ π , f (P ) = P (i = 1, 2, 3), 80

1

2

1

2

1

2

1

3

2

3

i

i

0 # > 0 2.29 ϕ : π −→ π ϕ

/ f (P ) = P .

! 2.27

35

P , P , P , P π "(% ϕ(P ) = P (i = 1, 2, 3), ϕ(P ) = P . P , P , P , P π "($ 2.28, / f : π −→ π , f (P ) = P (i = 1, 2, 3), f (P ) = P . 0 ϕ = f . X π % ϕ(X) = X , f (X) = X . 0 X = X . 2.27, P P P P E , ϕ(E ) = E . E P P P P % 2.26 f (E ) = E . P P 1

2

3

i

i

1

2

3

i

i

1

1

2 3

1

2 3

1

1

1

1

1

1

1<#"-6 81 P X X , ϕ(X ) = X . X P P P X f (X ) = X . X P P P X Steiner 2.26, §2.6

3

1

1

X

1

1

3

1

1

3

1

1

1

(P1 P , X1 E1 ) = (P1 P, X1 E1 ) = (P1 P , X1 E1 ).

Æ X P X X P X Æ X = X . 2.15 " π π

/ (2.21) " π π -.

2.15 ! (2.21) " (2.21) 5' 1

= X1 ,

3

2

ρ = 0,

|A| = 0

ρxi =

3

aij xj ,

i = 1, 2, 3,

(2.21 )

j=1

!



x1





 x1

          = A ρ  x2  ,  x2      x3 x3

|A| = 0,

ρ = 0.

(2.21 )

0(2.21) 5%" % &" (2.21) 15 5

;G←→ ;G ;G←→ ;G ρxi =

ϕ ϕ−1

σxi =

3

j=1 3 j=1

aij xj , (i = 1, 2, 3)

λui =

Aji xj , (i = 1, 2, 3)

µui =

3

j=1 3 j=1

Aij uj , (i = 1, 2, 3) aji uj , (i = 1, 2, 3)

A a 5$ (2.21) & ρ = 0, |A| = 0, 5 1& % λ = 0, σ = 0, µ = 0. 2.30 $ ( " ) ! $) $ '! , ! ! :6% ij

.

ij

82

1

1% #"-6

5 π P (1, 0, 0), P (0, 1, 0), P (0, 0, 1), P (1, 1, 1) 2 π P (1, 0, 1), P (0, 1, 1), P (1, 3

2

2.17

1

4

1

2

2

3

1, 1), P4 (0, 0, 1).

30 (&"5    ρx1 = a11 x1 + a12 x2 + a13 x3 ,    ρx2 = a21 x1 + a22 x2 + a23 x3 ,      ρx = a x + a x + a x , 31 1 32 2 33 3 3

2 ( P1 (1, 0, 0) ↔ P1 (1, 0, 1)

P2 (0, 1, 0) ↔ P2 (0, 1, 1)

P3 (0, 0, 1) ↔ P3 (1, 1, 1)

P4 (1, 1, 1) ↔ P4 (0, 0, 1)

|A| = |aij | = 0, ρ = 0.

   ρ1 = a11 ,    0 = a21 ,      ρ1 = a31 ;    0 = a12 ,    ρ2 = a22 ,      ρ =a ; 2 32    ρ3 = a13 ,    ρ3 = a23 ,      ρ =a ; 3 33    0 = a11 + a12 + a13 ,    0 = a21 + a22 + a23 ,      ρ =a +a +a . 4 31 32 33

(2.21)

12 5 ! ρ (Æ ρ a 13 12 (), a (D i

i

ij

ij

a11 : a12 : a13 : a21 : a22 : a23 : a31 : a32 : a33

J J − 1J0J1J − 1J1J1J − 1. #5  =1 0

  ρx1    ρx2      ρx 3

−x3 ,

= x1 = = x1

x2

−x3 ,

+x2

−x3 .

1<#"-6 83

- 2.16 &"!" - . 7 (2.21) 5 $ Æ , (x , x , x ) (x , x , x ) (2.21) 5Æ 5

&%1 1. P (y , y , y ) ρx = a x ⇐⇒ y : y : y = y : y : y ⇐⇒ α = 0, y = αy (i = 1, 2, 3) ⇐⇒ §2.6

1

1

2

3

i

⇐⇒

λ

j=1

ij j

1

2

2

1

3

3

2

i

ραyi =

% λ = ρα ⇐⇒

1

3

3

i

3

2

3

aij yj

(i = 1, 2, 3),

j=1

   (a11 − λ)y1 + a12 y2 + a13 y3 = 0,    a21 y1 + (a22 − λ)y2 + a23 y3 = 0,      a y + a y + (a − λ)y = 0. 31 1 32 2 33 3

a11 − λ a 21 a31

(2.22)

a23 = |A − λE| = f (λ) = 0, a33 − λ

a12

a13

a22 − λ a32

(2.23)

2.31 ϕ ⇐⇒ ϕ 1 A * R5 f (λ) = 0 2 J 4 2.14 J 2. l[v] µu = a u ⇐⇒ v : v : v = v : v : v ⇐⇒ κ = 0, v = κv (i = 1, 2, 3) ⇐⇒ 3

i

i

j=1

ji j

1

i

µκvi =

3 j=1

aji vj

(i = 1, 2, 3)

2

3

1

2

3

1% #"-6

84

! $ ), % γ = µκ ⇐⇒

(

⇐⇒

   (a11 − γ)v1 + a21 v2 + a31 v3 = 0,    a12 v1 + (a22 − γ)v2 + a32 v3 = 0,      a v + a v + (a − γ)v = 0 13 1 23 2 33 3

γ a11 − γ a12 a13

a21 a22 − γ a23

a32 = |A − γE| = |(A − γE) | = f (γ) = 0. a33 − γ

(2.22 )

a31

(2.23 )

2.31 ϕ ⇐⇒ ϕ 1 A * R5 f (λ) f (γ) R5 4 2.14 J 1 2.18

   ρx1 = −x1    x1 ρx2 =      ρx = x1 3

&% 2 () ** (1) 1*

+x2 + x3 , −x2 + x3 ,

(1)

+x2 − x3

−1 − λ 1 1 f (λ) = 1 −1 − λ 1 1 1 −1 − λ

= (λ + 2)2 (λ − 1) = 0,

* λ = −2( ), λ = 1(6). ( ) (1) ( 1

2

   (−1 − λ)x1 + x2 + x3 = 0,    x1 + (−1 − λ)x2 + x3 = 0,      x + x + (−1 − λ)x = 0. 1 2 3

(2)

1<#"-6 λ = −2 ( (2), (/ 1, 7

§2.6

85

1

x1 + x2 + x3 = 0.

# λ = −2 & (). λ = 1 ( (2) 1

2

   −2x1 + x2 + x3 = 0,    x1 − 2x2 + x3 = 0,      x + x − 2x = 0. 1 2 3

(/ 2, 7

   −2x1 + x2 + x3 = 0,  

x1 − 2x2 + x3 = 0.

λ = 1 (1, 1, 1). () (1) ( 2

   (−1 − λ)u1 + u2 + u3 = 0,    u1 + (−1 − λ)u2 + u3 = 0,      u + u + (−1 − λ)u = 0. 1 2 3

λ

1

= −2

(2 )

( (2 ), (/ 1, 7

u1 + u2 + u3 = 0.

(1, 1, 1), # λ = −2 ( ( (( ). λ = 1 ( (2 ) 1

2

   −2u1 + u2 + u3 = 0,    u1 − 2u2 + u3 = 0,      u + u − 2u = 0. 1 2 3

1% #"-6

86

(/ 2, 7

   −2u1 + u2 + u3 = 0,  

λ

2

=1

u1 − 2u2 + u3 = 0.

[1, 1, 1], # x

1

. /

2.6

+ x2 + x3 = 0

/*H π +/0* A(0, 0, 1), B(1, 2, 3), C(2, −1, 4), D(−1, 0, 1) 8%/.0* A (1, 4, 5), B (−1, 13, 14), C (11, 23, 28), D (−1, 1, 2). ,#"-6B) 2. /*H π +/0* (0, 0, 1), (1, 0, 0), (−1, 1, 1), (1, 1, 0) 8%/.0* (1, −1, 1), (2, 1, 0), (−2, 0, 1), (1, 3, 0). ,#"-6B(-,R/T<)/Q)H+)#"-6B) 3. /3#"-6      1.

x  1  ρ  x2  x3

−2

1     = 4   1

−2 −1

x1   3   x2  0 x3 1

  . 

,* P (1, 3, 2), Q(−1, 2, 5), R(0, 4, −3) //#"-6/)+*) - 1) 4. ,#"-6  )#->V) 5. ,#"-6

)#->V) 6. ,#"-6

)#->V)

 ρx1     ρx2      ρx3

  ρx1     ρx2      ρx3

  ρx1     ρx2      ρx 3

=

x1

=

+x2 , x2 ,

=

x3

−x3 ,

=

x1

=

x1

+2x2

+x3 ,

=

2x1

+2x2

+3x3

=

7x1

+4x2

−x3 ,

=

−x1

+2x2

+x3 ,

=

−2x1

−2x2

+5x3

1<#"-6 7. ,#"-6

§2.6

)#->V) 8. ,#"-6 )#->V)

87   ρx1     ρx2      ρx3

  ρx1     ρx2      ρx3

=

2x1

+2x2

+x3 ,

=

x1

+3x2

+x3 ,

=

x1

+2x2

+2x3

=

2x1 ,

=

6x1

+2x2

−x3 ,

=

3x3

G A=HUMLJ >?) 7 (:6 ( ) GC % Æ Klein(

§3.1

.

PJ9:KJ2NE

:6"&

3.1

$Æ x & 3.1 ϕ

=0

"&

|A| = |aij | = 0, ρ = 0,

(3.1)

3

   ρx1 = a11 x1 + a12 x2 + a13 x3 ,    ρx2 = a21 x1 + a22 x2 + a23 x3 ,      ρx = a x + a x + a x , 31 1 32 2 33 3 3

:6 x

35

a31 x1 + a32

⇐⇒ (3.1) a = a = 0. =⇒ ϕ x = 0 x = 0 (3.1) x ≡ 0, Æ a = a = 0. (3.1) 5 3

=0

31

32

3

3

2

31

32

   ρx1 = a11 x1 + a12 x2 + a13 x3 ,    ρx2 = a21 x1 + a22 x2 + a23 x3 ,      ρx = a33 x3 , 3

⇐= (3.1) 5 "% a = a 0@ (3.2) 5 |A| = 0, Æ a A 31

32

33

= 0.

33

88

= 0.

|A| = |aij | = 0, ρ = 0.

(3.2)

(3.1) (3.2), :6 x

3

a

33

= 0, A33

a11 = a21

=0

a12 = a22

1<#"-6)OF 0. (3.2) BN! ρx a §3.1

3

  x = a1 x + b1 y + c1 ,  y = a x + b y + c , 2 2 2

89 33 x3 ,

"

b1 = 0. b2

a1 a2

(3.3)

(3.2) 5" (3.3) 5 (Æ ) "& $ (3.3) 5% % $ N"# ( 2 $ , (3.3)

.

& Æ . >>

%

RM 3.2 3( a11 a21

   x = a11 x + a12 y + a13 ,   y = a21 x + a22 y + a23 ,

a12 = 0 a22

(3.4)





 1Æ AA = A A = E, $ 1A= a a RM . (3.4) 5 (3.2) RM . 1(@ (3.4) 5  a11

a12

21

22

  x = x cos θ − λy sin θ + a,   y = x sin θ + λy cos θ + b

(λ = ±1),

(3.5)

a, b, θ * 62 DÆN!)-,

. /

3.1

/34<+!&0 x + y = r (9."(I#-6 (3.3) $)(5)+2GM2 0.=#2(523!!,. 2. ,0 x + y = r /%0-6 (3.4) * (3.5) /)+).+0/%0-6/)+2GM 20. 2

1.

2

2

2

2

2

/% -6I./N 3. #"%0-6MI635!)+)&$ E ;* I(1,i,0),J(1,-i,0) #-(;&*9* SYZ. "-++43)0?"(&0O*) 4. / -62G2I#-6., $/&2%0-6.   90

  x = x + 2y + 3,

(1)

(3)

(5)

(2)

  y = x + 3y + 2.  1 √   x = ( 3x − y + 1), 2 √ 1   y = (x + 3y + 3). 2    x = 3x − 4y + 1,   y = 4x + 3y − 1.

§3.2

1   x = (3x + 4y + 5), 5   y = 1 (4x − 3y − 5). 5    x = x,

(4)

  y = −y.    x = 2x + 1,

(6)

  y = 2y − 1.

12NJKJJ

$ 2J ,

J" 3.3

G

G

G

G.

" ∀a, b ∈ G ab ∈ G. # J -? ∀a, b, c ∈ G a(bc) = (ab)c. $ 6& e ∈ G ∀a ∈ G ae = ea = a. % &%0&% ∀a ∈ G a ∈ G aa = e = a a. $D%J"%% 1 {1} D%J "+ n O01 1J" ! 3.2 J( G -" 2ÆJ % K J2 ?" Abel $ 3.2 ÆGP25Æ 3.4 G H ⊆ G, H = φ. " H G J2" H G T. 3.2 H ⊆ G. H G ⇐⇒ (1) " h , h ∈ H h h ∈ H; (2) " h ∈ H, h ∈ H. −1

1

2

1 2

−1

−1

−1

+)/&-6I 91 3.5 G, G "$ f : G → G ∀a,∼b ∈ G, & f (ab) = f (a)f (b) = a b ∈ G . G G UK G = G . 3.3 S J" S F . ϕ , ϕ J-, + J 3.3 3.4 G S G J " ⇐⇒ (1) " g , g ∈ G g g ∈ G; (2) " g ∈ G, g ∈ G. G 3.6 G S G S .

[\ 1. 3.5 K J" K . 35 30 K J 3.4. (1) κ , κ ∈ K. §3.2

1

1

2

2

1 2

−1

1

2



κ1 :

x1





x1

        ρ  x2  = A  x2  ,     x3 x3 

κ2 :



x1





|A| = 0, ρ = 0,

 x1

        σ  x2  = B  x2  ,     x3 x3

|B| = 0, σ = 0.

X κ X , X κ X , $) X κ , κ J X .

1

2

1



x1

  ρσ  x2  x3



 

x1







x1

             = B ρ  x2  = (BA)  x2  .       x3 x3

2

/% -6I./N % ρσ = λ, BA = C, λ = 0, |C| = 0. Æ κ κ J 92

1

 κ:

x1





2

 x1

        λ  x2  = C  x2  ,     x3 x3

|C| = 0, λ = 0

2Æ κ ∈ K. K &%J K (2) / #0%02 Æ κ ∈ K, κ ∈ K. - K 2. " 3.5, 0 3.6 KA " ; A " A −1

.

KA A "#

3.7. 3.7 KA K % KA ∼= A. 3. > - 3.8 KM " R M ; M " M RM ∼. 3.9 KM KA % KM = M.

- P P A P :

K

⊃ KA ⊃

KM

∼ =

∼ =

PA :

A

. /

1.

9E++, 3.6, 3.7, 3.8 . 3.9 )"-)

3.2

⊃

M

§3.3

-6I./N

93

KJJQNPK F. Klein RQ F. Klein 1872 RQ Erlangen 3W E E F. Klein ! Erlangen KH 4& @ K2 XJ S ' Erlangen KH % ! KH O &WQ F. Klein ] 3.7 S G S S => S &% S F G S V . S ' G &% :6 ( G ) ( G ) 7 (S, G). $ 3.7, S G, "7! F, F S (Æ S ), " g ∈ G, g(F ) = F . F F " F ∼ F . 0 ∼ S " "Æ (1) . F ∼ F , ." g = e G 6 & (Æ2) Æ (2) " F ∼ F , F ∼ F . G &% G 0&% G F F g, F F 0 g ; (3) DN" F ∼ F , F ∼ F , F ∼ F . J Æ" g, f F F F F G f g, F F . $ G S " ( ""(" " · · · · · · ), " S 3.8 (S, G) H G (S, H) §3.3

−1

/% -6I./N

94

(S, G) ^T , T . Σ ⊂ S, Σ = φ, H G % g ∈ H, g(Σ) = Σ, H Σ % H ∼ = H (Σ, H ) (S, G) (S, H) _W^T QT , B = S \ Σ (Σ, H ) ^+. KA ⊆ K, τ ∈ KA, τ (P \ l ) = P \ l , ) Σ = P \ l = P A, A ∼= KA, (P A, A) (P, K) (P, KA) N A l A G H G :6$ H G G (S, H) LLL+:6

) 2 LM$) :LM $KH -

Σ

Σ

Σ

Σ

∞

∞

∞

∞

(P, K)

⇐=

(P, KA)

⇐=

(P, KM )

(P A, A)

⇐=

(P A, M )

⇐= A ←− = N A& l .

[X N` 1. (&% ) $ % (Æ2 2) (Æ (((%( ), $ EL ( 2. ∞

-6I./N 95 !"& :6#"& A )" $"& 3% .7 " :6"& ) "& "& #"& ( ( ( 3.10 :6

(Æ ) C:6 "& "& $ " "-, PM (" " 3NY 6( ":6:6

+6 3.9 P , P % %P % P P §3.3

1

2

(P1 P2 P ) =

1

P , P , P a;N ;N. P , P P $. 4 3.1 P , P , P % l % P l "& 1

2

1

1

35

P2 P 2

2

∞

(P1 P2 P ) = (P1 P2 , P P∞ ).

6 P , P , P l % P l "& l l , l % 2% P , P , P ; P l "& 1

3.11

1

2

∞

2

∞

(P1 P2 P ) = (P1 P2 , P P∞ ) = (P1 P2 , P P∞ ) = (P1 P2 P ).

0@ V 6 $EL EL ( (H !

/% -6I./N

96

3%Æ 3.12 35 P (x , y ), P (x , y ) (3.4) P (x , y ), P (x , y ). $ (3.4), 3.

1

1

1

1

1

1

2

2

2

2

 

 

x1 y1 x2 y2

2

2







 = A

x1

+

y1 

 = A

x2

a13

,

(3.6)

a23 











+

y2

a13

,

(3.7)

a23

A 1Æ AA = A A = E. (3.7) (3.6),

 

x2 − x1 y2 − y1





 = A

x2 − x1 y2 − y1

 .

(3.8)

(3.8) N (x2 − x1 , y2 − y1 ) = (x2 − x1 , y2 − y1 )A .

(3.9)

(3.9) !BJ (3.8) ! A A = E

(x2 − x1 )2 + (y2 − y1 )2 = (x2 − x1 )2 + (y2 − y1 )2 .

5):60@ ÆEL Æ ( ! % :6 -$ $ $2$

. /

1.

3.3

/ !,-2N9/N (&B3R(?)-6I) )K3/T.

-6I./N 97 (1) -,/ (2) /*,)K(/ (3) -0,,/ (4) 0/ (5) /*,) $(/ (6) I.* / (7) )M)$!%)/ (8) 8<0*,/ (9) &AM )/*,/ (10) %.,) 2. / /N5-2N9/N (&B3R(?)-6I) )K3/T. (1) )M)MS/ (2) !))S*/ (3) /*,)R/ (4) *(!))C ;/ (5) -)8)C;) 3. / /NOP-2N9/N (&B3R(?)-6I) )K3/T. (1) -/ (2) $!/ (3) -0,,)/*)/7%/ (4) $)**$* )) 4. ,"+&/*,)R$C*I##-5)<9"--5&OO!,)R)C2 I##-5) §3.3

! 3.1 5. %.,PI,)$0OP2I##-). 6. +

OP = x e1 +y e2 .

T O e e 78R O . e , e A+) bcYdZ, 9 (x, y) ** P / O e e /) bcYd(. ! 3.1). "-+  1 2

1 2

1 2

OPx = (P E O),   x = OE x x x   y = OPy = (Py Ey O). OEy

<9(/#"-19 (A A A |E), / A A *635!)(9Z-I#- 192#"- 19 )OF) 1

2

3

1

2

[ HNLUDJ §4.1

J7

P&MM29:FN

4.1

S≡

3

T ≡

aij ui uj = 0, (aij = aji )

i,j=1

4.1

3

bij xi xj = 0, (bij = bji ) (4.1 )

i,j=1

(4.1)

(x , x , x ) [u , u , u ] . B = (b )

e. A = (a ) O f 1/ (B) 1. 1/ (A) 1. O 5& $$ O . " 5 O 1

2

1

3

2

3

ij

ij





a11

  S ≡ (x1 , x2 , x3 )  a12  a13

a12

a13

a22

a23

a23

a33



x1

      x2  = 0,   x3

/(a

ij )

1.

" X = (x , x , x ), 1

2

3

/(A) 1. 4.2 " 4.1 4.2 " 4.1 5 S 5J 5 T 5J O S = 0 [g T = 0 [g , 0[g , 0[g Q" ORQ" R 0 S = 0 Q" ⇐⇒ |a | = 0; T = 0 Q" ⇐⇒ |b | = 0. S ≡ XAX = 0,

A = A ,

ij

ij

98

1%0))#"+< 99

4.1 4.1 & (" " (( O % & 35 0 4.1. O(p)∧O (p ). ( O(p) §4.1

A ≡ a1 x1 + a2 x2 + a3 x3 = 0,

l1 :

l2 :

B ≡ b1 x1 + b2 x2 + b3 x3 = 0.

O(p) A + λB = 0(λ ∈ R). ( O (p ) A + λ B = 0(λ ∈ R). O(p)∧O (p ),

aλλ + bλ + cλ + d = 0,

λ, λ

$(!

ad − bc = 0.

   A + Bλ = 0,    A + B λ = 0,      aλλ + bλ + cλ + d = 0. λ, λ ,

! 4.1 aAA + dBB − bAB − cA B = 0.

(4.2)

(4.2) 5 x , x , x (4.1), O   1

O:

2

3

 A = 0,  B = 0,

O :

 A = 0,  B = 0

0% 1%0),3 O, O (4.2), Æ O, O O (4.2) 0@ 4.2 O Γ $ 4.2 Γ $ ( O(p)∧O (p ) Γ l(P )∧l (P ) Γ A, B Γ a, b Γ M ( A(AM )∧ m a(a × 100

35

m)∧b(b × m).

B(BM ).

-0 4.2. O Γ $ O(p)∧O (p ) 4.2 O(P )∧O (P ). A, B Γ M Γ % AM × OP = K, BM × O P = K .

A(M )∧OP (K),

B(M )∧O P (K ).

O A BM A , OB AM B . O(P )∧O (P ),

O(A, B, P, M )∧O (A, B, P, M ).

AM, BM $( AM (A, B , K, M )∧BM (A , B, K , M ).

&% M (A, B , K, M )∧(A , B, K , M ).

AA , BB , KK S. S = AA × BB = AO × BO M KK S, Æ OP (K)∧O P (K ). A(M )∧B(M ). 0@

! 4.2 4 4.1 ( " 4 4.1 ( ) !Q" O ") !Q"

1%0))#"+< 101 35 O, O , P , P , P & O, O ( P ( O(P ) O (P ). %! O(P )∧O (P ). $ 4.1, Q" O 4.2, O 2 1 A = (a ) 1 6 &%$ 6 &% 5 7 ,2% & !Q" 4 4.2 O$ 4 4.2 $ ( 4 4.3 O 4 4.3 2 4.3 4.3 ( O Steiner " $ 3%Q" 1 4.1 $( x − λx = 0, x − µx = 0(λ + µ = 1) O 2h % §4.1

i

1

2

3

i

i

ij

1

3

2

3

A ≡ x1 = 0,

B ≡ −x3 = 0,

A ≡ x2 = 0,

B ≡ −x3 = 0,

λ + µ = 1 *5(    A + λB = 0,

  A + µB = 0.

$ 4.1 0 ( O aAA + dBB − bAB − cA B = 0.

a = 0, b = c = 1, d = −1 ( (1), x1 x3 + x2 x3 − x3 = 0,

(1)

0% 1%0),3

102

Æ x3 = 0, x1 + x2 − x3 = 0.

Q" O 2h $ (x , x , x ) ( ⇐⇒ 1

2

3

   x1 − λx3 = 0,

  x2 − (1 − λ)x3 = 0.

"

   x1 − λx3 = 0,   (x2 − x3 ) + λx3 = 0.

(x , x , x ) ( ⇐⇒ ( (1, λ) $Æ 1

2

3

x1 x3 + x2 x3 − x3 = 0.

1 !% ' D'& % ' 2 ! 4.2

(

ABC

)

AB, BC, CA

P, Q, R,

a, b D A B C , A B C , · · · D ( 4.3). $

A, B

1

1

1

2

! 4.3 R(C1 , C2 , · · · )∧P (A1 , A2 , · · · ), Q(C1 , C2 , · · · )∧P (B1 , B2 , · · · ).

2

2

1%0))#"+<

§4.1

103

R(C1 , C2 , · · · )∧Q(C1 , C2 , · · · ).

$ 4.3 ' C & Q, R O

ei fi Æ Q" 1. O* 4.4 O Γ Γ i.

O* O 3

Γ : S≡

aij xi xj = 0,

|(aij )| = 0.

(aij = aji ),

(4.3)

i,j=1

P Γ * Q(q ) P Q x = p + λq (i = 1, 2, 3). P Q Γ * ⇐⇒ x = p + λq ( (4.3) ( P (pi )

i

i

i

i

i

i

i

3

aij (pi + λqi )(pj + λqj ) = 0,

i,j=1

Æ

3 i,j=1

I λ2

aij (pi pj + λpi qj + λqi pj + λ2 qi qj ) = 0,

3

 aij qi qj + λ 

i,j=1

3

aij pi qj +

i,j=1

3



3

aij qi pj  +

i,j=1

aij pi pj = 0.

(4.4)

i,j=1

(

 3   Spp ≡ aij pi pj ,    i,j=1    3 Spq ≡ aij pi qj ,  i,j=1    3    aij pi xj ,  Sp ≡ i,j=1

a

ij

= aji ,

S

pq

= Sqp .

Sqq ≡ Sqp ≡ Sq ≡

3 i,j=1 3 i,j=1 3 i,j=1

aij qi qj , aij qi pj ,

(4.5)

aij qi xj ,

( (4.4),

Sqq λ2 + 2Spq λ + Spp = 0.

(4.6)

0% 1%0),3 Q Γ P * ⇐⇒ (4.6) λ Æ 104

2 Spq = Sqq Spp .

(4.7)

Æ Q(q ) Γ P *(&5' q x Γ P * (4.7)

i

i

i

Sp2 = Spp S.

" P Γ (4.8) S

pp

= 0,

(4.8)

Γ P *

Sp = 0.

(4.9)

(4.9) 5 F5@

0Æ

∂S ∂x1

! 1 5

p

∂S ∂x1

x1 +

∂S ∂x2

p1 +

∂S ∂x2

p

x2 +

p2 +

∂S ∂x3

∂S ∂x3

p

a11

  (p1 , p2 , p3 )  a12  a13

a12

a13

a22

a23

a23

a33

(4.10)

p3 = 0,





x3 = 0,

(4.11)

 x1

      x2  = 0.   x3

(4.12)

* O* * 2.

Γ : T ≡

3

bij ui uj = 0, (bij = bji ), |(bij )| = 0.

(4.3 )

i,j=1

" % Γ

i. O P *$ l[l ] * m[m ] l l × m, , l × m *5 ( l × m ξ = l + λm , (i = 1, 2, 3), ( Γ Γ

4.4

i

i

i

Tmm λ2 + 2Tlm λ + Tll = 0.

i

i

(4.6 )

1%0))#"+< 105

! O l × m Γ * ⇐⇒ λ ⇐⇒ §4.1

2 Tlm = Tll Tmm .

(4.7 )

Tl2 = Tll T.

(4.8 )

) m u , 5 i

i

2 l * , l ∈ Γ ,

* T= 0,Æ

l

∂T ∂u1

l

∂T ∂u2

u1 +

l

∂T ∂u3

u2 +

l

(4.10 )

u3 = 0.

*$ 5

e f\ O #+ 4.3(Maclaurin ) Q" O*" Q" Q" *"Q" O 35 Q" O Γ : S≡

3

aij xi xj = 0

|(aij )| = 0.

(aij = aji ),

i,j=1

Γ P (p ) * [u, u , u ]Æ u x + u x + ux = 0. Γ ∂S ∂S ∂S P * S = 0 Æ x + x + x = 0, # ∂x ∂x ∂x i

1

p

Æ Æ

∂S ∂x1 u1

p

1

=

2

p

3

1 1

1

∂S ∂x2 u2

2

p

=

2 2

2

p

∂S ∂x3 u3

3 3

3

p

p

3

),

= ρ(

a12 p1 + a22 p2 + a23 p3 a13 p1 + a23 p2 + a33 p3 a11 p1 + a12 p2 + a13 p3 = = = ρ, u1 u2 u3    a11 p1 + a12 p2 + a13 p3        a12 p1 + a22 p2 + a23 p3   a13 p1 + a23 p2 + a33 p3        u 1 p1 + u 2 p2 + u 3 p3

−ρu1 = 0, −ρu2 = 0, −ρu3 = 0, = 0.

0% 1%0),3 [u , u , u ] Γ P *(&( (p , p , p , ρ) $Æ 106

1

2

3

1

a11

a12

a13

a12

a22

a23

a13

a23

a33

u1

u2

u3

u1 u2 = 0. u3 0

2

3

(4.13)

T ≡

3

%A

Aij ui uj = 0,

ij

i,j=1

= Aji , |(Aij )| = |(aij )|2 = 0.

T = 0 Q" &0$Q"

Γ : T ≡

3

bij ui uj = 0

(bij = bji ),

|(bij )| = 0.

i,j=1

(4.13) Æ S ≡

3

%B

Bij xi xj = 0,

ij

i,j=1

S

= Bji , |(Bij )| = |(bij )|2 = 0.

Q" O 4 4.4 " 4.3 b = αA (α = 0), S = 0 T = 0 4.3 0Q" " e, O(:: O O

=0

ij

Γ1 : f ≡

3

ij

aij xi xj = 0

(aij = aji ),

bij xi xj = 0

(bij = bji ),

i,j=1

Γ2 : g ≡

3 i,j=1

§4.1

1%0))#"+<

107

  f = 0,  g=0

Γ Γ (). 3L(2 4 (Æ Γ Γ 2 4.4 O2 4.5 f = 0, g = 0 O$ 1

2

1

2

λ∈R

f + λg = 0,

(4.14)

' O f = 0, g = 0 e,. , f = 0, g = 0 O( e +,. 4.5 P (), O"

O( f + λg = 0. 35 P (p ) f = 0 g = 0 f g $ ) g = 0. % i

pp

pp

λ=−

f+

f − gpp g=0 pp

pp

fpp , gpp

f = 0 g = 0 P O0@ 4.6 O( Q" O '$!

! 4.4

> 0 4.6

ABCD, A, B, C, D O ( Q" O Γ10 : AB · CD = 0,

0% 1%0),3

108

Γ20 : BC · AD = 0, Γ30 : AC · BD = 0.

$ O(* Q" O OU 1 4.3 A, B, C, D, E( ") $ ! O 2 O( 2 () A, B, C, D, $! O ( O( Q" O ( ), f = AB · CD g = AC · BD, O(* E (

f + λg = 0. ( )

λ=−

fee , gee

( f + λg = 0 Æ O 1 4.4 O Γ A(1, 0, 1), C(0, 0, 1), E(3, 2, 1), % Γ * x − 3x − x = 0, 2x − x = 0. Γ 2 A x − 3x − x = 0 C 2x − x = 0 Γ

* A, C( 4.5). O( % A ≡ B, C ≡ D, O( O( Q" O 1

1

2

2

3

1

2

3

1

2

! 4.5 AB : x1 − 3x2 − x3 = 0,

CD : 2x1 − x2 = 0,

AC : x2 = 0,

BD : x2 = 0.

A, B, C, D O(*

AB · CD + λAC · BD = 0,

1%0))#"+<

§4.1

Æ

109

(x1 − 3x2 − x3 )(2x1 − x2 ) + λx22 = 0.

E(3, 2, 1) (5 λ = 2. # O

1

2x21 + 7x22 − 7x1 x2 − 2x1 x3 + x2 x3 = 0.

P (p , p , p ) J O Γ : S = 0 ** H , K ,

Q(q , q , q ) J S = 0 ** H , K . 0 H , K , P, H , K , Q O 35 P O Γ P Γ * 1

4.5

2

1

2

3

1

3

2

1

1

2

1

2

2

Spp · S = Sp2 ,

* H , K (  1

1

  Spp · S = Sp2 ,

⇒

  S = 0,

#S

  S = 0,

H K * H , K  ( 

p

=0

1

2

1

2

  Sqq · S = Sq2 ,

⇒

  S = 0,

#S

   Sp = 0,

q

=0

  Sq = 0,   S = 0,

H K 2

2

! 4.6 ! O H , K , H , K Γ + H K , H K Q" O H , K , H , K O ( 1

1

1

2

1

2

2

2

1

Sp · Sq + λS = 0.

1

2

2

(∗)

0% 1%0),3

110

P (p , p , p ) ( (∗) ! λ 1

2

3

λ = −Sqp (= −Spq ),

# H , K , P, H , K O 1

1

2

2

Sp · Sq − Spq · S = 0.

30 Q(q , q , q ) 0@ 4.6 Q" O O O ( Q" O , O( Q" O 2 $ O( Q" O \ "D 1

2

3

. /

4.1

,6(Æ* (1, 0, −1), (1, 0, 1), (1, 2, 1), (1, 2, −1), (1, 3, 0) )1V0)..) 2. ,6(Æ* (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1), (a , a , a ) )1V0)..) 3. ,6(* (1, 0, 1), (0, 1, 1), (0, −1, 1), 5- x − x = 0 . x − x = 0 *N))1V 0)..) 4. ,R/ #")4]1)1V0)..+ (1) x − x − x + λx = 0 . x + 2x − x + λ (x − 4x + x ) = 0, 5 λ − λ = 0; (2) x +2x +λ(x −x ) = 0 . x +x +x +λ (x −2x ) = 0, 5 2λλ +λ−2λ +3 = 0. 5. .! 4.7, //*, ABC . A B C /1EN!,1%0)),"5,B/1#W !,1%0)) 6. "-+.)6>.1K) 1.

1

1

1 1

2

3

2

3

1

1

3

2

1

2

3

3

3

3

2

2

2

1

2

3

3

3

! 4.7 7. /74,1%0) Γ . Γ -C#W Γ -/1EN Γ )!&/*,)9K32 G,/,F/*,B;?/,2(=,/(3OO. 8. /* P (p , p , p ) /1V0) Γ : x − 2px x = 0 +),6( P ) Γ )N)..)

1

2

3

2 2

1 3

+,. Brianchon +, 111 9. /E+1X0) u + u − 17u = 0 .!) [1, 4, 1]. ,0)//!)+)N*) 10. ,"+/ )* - 1...) - 1..78/!,0)) 2u u ; (1) x = 2px x . u = p x x (2) + = x . au + bu = u ; a b (3) x x − x = 0 . 4u u − u = 0; (4) x x = c x (c = 0) . 4c u u = u .

§4.2 Pascal

2 1

2 2 2 1 2

1 3

2 3

1 3

2 2

1 2

2 2 3

2 3

2 2

1 3

2 2 2

2 2

2 1

2 2

2 3

2 2

1 3

2

2 3

1 2

F7D Brianchon F7 + ?+0 - 68 * 68 68 O'" 68 *68 !"68 -68!68'$ 68!68 A A A A A A , $ H A A A A A A A A A A A A . Pascal P Brianchon 4.7(Pascal ) 4.7 (Brianchon ) * Q" O Γ 68$ Q" Γ 68 ! Pascal . $' Brianchon . 35 0 4.7. 4.8, 68 A A A A A A Q" O Γ . A A × A A = L, A A × A A = M , A A × A A = N . A , A -$2 4.3 §4.2

Pascal

1

1

2

4

5

2

3

5

6

2

3

3

4

4

5

6

6

1

1

1

1

2

4

5

2

3

5

6

3

! 4.8 A1 (A2 , A4 , A5 , A6 )∧A3 (A2 , A4 , A5 , A6 ).

2

3

3

4

4

5

6

6

1

0% 1%0),3

112

A A 1

6

× A4 A5 = P , A3 A4 × A5 A6 = Q.

A1 (A2 , A4 , A5 , A6 )∧(L, A4 , A5 , P ),

A3 (A2 , A4 , A5 , A6 )∧(M, Q, A5 , A6 ).

(L, A4 , A5 , P )∧(M, Q, A5 , A6 ).

A 5

(L, A4 , A5 , P )∧(M, Q, A5 , A6 ).

LM, A Q, P A Æ L, M, N 0 1 " L, M, N 0 24 Æ 4

6

A1 (A2 , A4 , A5 , A6 )∧A3 (A2 , A4 , A5 , A6 ).

)688' O0 4.8 "68 4.8 "68 $!8 $'8 * O Γ . Γ . 2 4.8 O 1 4.6 A, B, C, D, E( "). $ ! O Γ F . 2 (1) - AB, DE L. L p( ). (2) - BC p M , - CD p N . (3) - EM AN F

! 4.9

+,. Brianchon +, 113 0

!68 ABCDEF , $! L, M, N p, Pascal 0 4.8, 8 ABCDEF O Γ . p, Æ Γ

Pascal j]+=

% Pascal 5$ Brianchon 55 O 68(L' L'- '! O * 1. L' 4.9 Q" O Γ 6(*! L! §4.2 Pascal

! 4.10

! 4.11

L' 4.10 Q" O6(!(' * 4.11 Q" O6!+ ! '* 2.

! 4.12

! 4.13

0% 1%0),3

114

L' 4.12 Q" O '*! OQ"688'2 9 Pascal Q" Pappus Pappus Pascal Q" 4.9 O * 1 4.7 O Γ Γ P Γ P * (1) Γ P A , A , A , A ; (2) A A A P L A A A P N (3) LN A A M (4) P M Æ P * ( 4.14). 35 (>). 3.

1

1

2

4

2

3

4

2

3

4

1

3

! 4.14

! 4.15

Brianchon 01 1 4.8 ( ") % * O

A, B, C, D(") C l, $& !Q" O Γ A, B, C, D % l * C. 2 Pascal 50 C X 8 , &8 DABCCX, $! Pascal

DA × CC(l) = P,

4.15 (1)

AD × l = P ;

AB × CX = Q,

BC × DX = R,

+,. Brianchon +, P Pascal p(= l);

§4.2 Pascal

(2) (3)

115

AB × p = Q, BC × p = R;

CQ × DR = X 0 Pascal 5 () 0 P, Q, R X A, B, C, D % l * C O$ ! Pascal p 1 4.9 4.16, ABC A B C * (4)

AB × A B = X,

BC × B C = Y,

CA × C A = Z.

0 X, Y, Z 35 !* 68 ABCA B C , X, Y, Z F$! Pascal X, Y, Z

1

! 4.16

! 4.17

* ABCD A B C D , AB × A B = P, = Q, CD × C D = R DA × D A = S

4.10

BC × B C

( 4.17). 35 *Q" O & 68 ABCA B C $ ! AB × A B = P,

BC × B C = Q,

CA × C A = T.

$ Pascal P, Q, T & 68 ADCA D C S, R, T P, Q, R P, Q, R, S

116

).

1.

. /

0% 1%0),3 4.2

9K3+/1V0)+4+)"&*(Q,#/)%O(-,(OO,#/) Pascal

%*"-+, 4.9∼ +, 4.12. 3. %*4++, 4.9∼ +, 4.12 )/8>.) 4. /$)/*, ABC . A B C 2AM) (.! 4.18). ,"+"!) AB , AC , BC , BA , CA , CB A/!,1X0)) 5. / A, B /1V0) Γ +( C, D #/ Γ +( AC, BD %*0 Γ P, Q; AD, BC % *0 Γ U, V (.! 4.19). ,"+ CD, P Q, U V /!)$*) 2.

! 4.18

! 4.19

/ A, B, C, D *1V0) Γ +)0&+*( P, Q * Γ +).*( P A × DC = X, P B × QD = Y . ,"+ XY (+*) (G k +8.MN..'O3(G?B k *3!N.) 6.

§4.3

H Y K J

($ ——– + O$ >! EF $ + Q" j j 4.6 , O Γ Γ C?. 4.13 P (p ) O Γ Æ& p, S = 0. i

p

IZ-6 35 O

§4.3

117

3

S≡

Γ :

(aij = aji ), |(aij )| = 0,

aij xi xj = 0

i,j=1

P (p , p , p ) Q(q , q , q ) + (Q ∈ Γ ). P Q 1

2

3

1

2

3

xi = pi + λqi PQ

Γ M (ξ

i

= pi + λqi )

(i = 1, 2, 3).

Sqq λ2 + 2Spq λ + Spp = 0.

(4.15)

P, Q & Γ # S = 0, S = 0, (4.15) ' P Q Γ M (p + λ q) (i = 1, 2). P, Q, M , M Æ ⇐⇒ λ /λ = −1, Æ λ + λ = 0, 0Æ S = 0. S = 0 P, Q O Γ Æ (& q x Æ P Γ Æ & p, S = 0. 4 4.5 P, Q O Γ Æ(& S = 0, Æ pp

qq

i

1

2

1

i

1

2

pq

i

2

pq

i

p

pq



a11

  (p1 , p2 , p3 )   a12  a13

a12

a13

a22

a23

a23

a33



   q1    q  = 0.  2    q3

(4.16)

P Γ O" P Γ P Γ Æ & p P &6 Γ j; " P Γ Γ P * p P &6 Γ j, P p &6 Γ j. $2 4.5, 4.6 "5 4.6 Γ Q" O Γ Æ 4 4.6 P O Γ S = 0. p Γ 35 0 &- u : u x + u x + u x = 0, P (p ) u P S = 0, Æ 4.7

p

1 1

2 2

i

3 3

p

∂S ∂x1

p

x1 +

∂S ∂x2

p

x2 +

∂S ∂x3

p

x3 = 0,

0% 1%0),3 u P # u S = 0 Æ 118

Æ

∂S ∂xi

p

p

ρ = 0, i = 1, 2, 3,

= ρui ,

  ρu1 = a11 p1 + a12 p2 + a13 p3 ,    ρu1 = a21 p1 + a22 p2 + a23 p3 ,     ρu = a p + a p + a p , 1 31 1 32 2 33 3

|aij | = 0.

(4.17)

( (4.17) j34I, &""/ Q"& |a | = 0 ) Γ $ 4.6 , 4.8 % O Γ C?. Maclaurin Æ p[p , p , p ], q[q , q , q ] Q" O Γ : S = 0 Æ ⇐⇒ ij

1





A12

A11

  (p1 , p2 , p3 )  A12  A13

A22 A23

2

3

1

2

3



q1

A13

    A23   q2  = 0,   A33 q3

(4.18)

A a (a ) 5 F Γ T ≡ 0 Æ & T = 0. 1 4.11 O Γ : ij

ij

ij

pq

3x21 + 5x22 + x23 + 7x1 x2 + 4x1 x3 + 5x2 x3 = 0.

(1)

2

(2) (1)

P (1, −1, 0) Γ p[0, 1, 0] Γ 30 O Γ Q" O Γ 



3

7/2



x1

2

      S ≡ (x1 , x2 , x3 )  7/2 5 5/2   x2  = 0,    2 5/2 1 x3

S

p

= 0,

Æ



 3

7/2

2

 x1

      (1, −1, 0)  7/2 5 5/2   x2  = 0,    x3 2 5/2 1

IZ-6

§4.3

Æx

1

119

(

+ 3x2 + x3 = 0.

(2)









3

0

7/2

 x1

2

          ρ  1  =  7/2 5 5/2   x2  ,      2 5/2 1 0 x3

(3, −2, −1). O" $ p[0, 1, 0] $

lj $Q" O Γ %& ""$ ? 4.9 $5 ρui =

3

aij xj .

aij = aji , (i = 1, 2, 3), |aij | = 0

(4.19)

j=1

'&"" O Γ : S ≡ a x x = 0 (|a | = 0) lj . % % , % . 4.14() P O Γ p Q ⇐⇒ Q Γ q P ; p Γ P q ⇐⇒ q Γ Q p 35 0& Γ : S = 0, P (p ), Q(q ). P Q ⇐⇒ S = 0. S = S , # S = 0 ⇐⇒ Q P . 4 4.7 35 A, B C, " C AB. C % A B, A B &% C. A, B C. 4 4.8 3

i,j=1

i

pq

pq

qp

qp

i

ij i j

ij

0% 1%0),3 35 A, B, C, · · · p, % p P . A ∈ p, P A $ A P , 0

- 4 4.9 Q" O ( 4 4.10 Q" O ! ( - $Q" O lj@A. 2 4.10 " 'Q" O Γ ! Γ j +. $ O ' O 4.15 'Q" O Γ ! Γ 35 4.20, ABCD (BC, EX) = −1, (AD, F X) = −1, E, F X Æ X Y Z. 0 Y XZ, Z XY . 1 4.12 4.21, ABC O Γ V RP % B RC + Q. 0 P Q % A. 120

3h

! 4.20

! 4.21

AC, RQ RB, AB S, T ; ABC (BS, RP ) = −1,

(T C, RQ) = −1.

IZ-6 §4.3

121

(B, S, R, P ) ∧ (T, C, R, Q).

& R =

(B, S, R, P ) ∧ (T, C, R, Q).

BT, SC, P Q BT SC Æ A # P Q % A. 3h - P C Γ M , C Γ P QM R ! RP × M Q = B (!). $ 4.15, C B , C AB, # B ∈ AB, $ B B AB RP Æ B ≡ B , M Q B. 1 4.13 P Q" O Γ P Γ p. 2 1. P Γ p Æ Γ P *1 "+ Pascal * 2. P Γ ^h (1) P Γ A, B, C, D( 4.22). (2) - BC, AD R, - AC, BD Q; (3) - QR Æ p.

35

! 4.22

4.15 4.22 QR Γ E, F . P E, P F Γ P * $ O *+ 1 4.14 p Q" O Γ , p Γ 2 P 0 1. p Γ * ^h p Γ A, B a, b, a, b P Æ

0% 1%0),3

122

2. p Γ * ^h p Γ A, 5 A Γ GF , GF p Æ p Γ (Æ*).

. /

4.3

P"/ *8274E/SS1V0))$ E*) (1) *8 (1, 0, 1), (6, 1, 0) 7 3x − 6x x + 5x − 4x x − 6x x + 10x = 0; (2) *8 (3, 3, 1), (3, 0, 1) 7 2x − 4x x + x − 2x x + 6x x − 3x = 0; (3) *8 (0, 3, 1), (1, 0, 1) 7 4x + 2x x − 6x x − 10x x + 15x = 0; (4) *8 (1, −1, 0), (1, 0, −1) 7 3x + 7x x + 5x + 4x x + 5x x + x = 0. 2. ,/ -*7E+1V0))Z)) (1) * (5, 1, 7) 7 2x + 3x + x − 6x x − 2x x − 4x x = 0; (2) * (6, 4, 1) 7 2x + 6x + 6x x − 2x x = 0. 3. ,/ !)7E+1V0))Z*) (1) !) 3x − x + 6x = 0 7 x − 2x x + x − 2x x + 6x x = 0; (2) !) x − x + 3x = 0 7 2x − 3x − 5x + 6x x + 3x x + 16x x = 0. 4. .! 4.23, ABCD 21V0) Γ )#W0*,( XY Z *,/,/*,),"+ B, C O)N). Y Z $*( A, D O)N)B. Y Z $*) 1.

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1

1

2 1

2 2

2 1

2 2

2

2

3

3

2 3

2 2

1 3

2 2

1 3

2 3

2 3

2 2

1 3

2 3

1 3

2 3

1 2

1 3

2 3

2 3

1 3

1 2

2 3

2 3

2 3

2 3

2 1

1 2

2 1

2 2

2 2

2 3

1 3

1 2

! 4.23

2 3

1 3

2 3

5. E+!,/SS1V0) Γ . "-+/#. Γ 7N)!)+363O27 Γ )$ E *8(5,91-/!)*Q)* +)!&/7)9.(//7)#->V23!. 6. //SS1V0) Γ +).* X, Y O)N)0*/!,+!)+)"-+.!) XY "(!&+*) 7. E+,1V0) Γ : Ax = 2x x , Γ : 2x x = Ax . 9,!*(:,7,1V 0))Z)*/!,!)) 8. , x + y = k ).N)7 ax + by = 1 )Z*)HI) 1

2

2

2

2 1

2 3

2

2

2

1 2

2 3

§4.4

1%0))#"%.

123

P&MM29:QG + O"Q" O O/ eQJ 4.11 " P (p ) ( §4.4

0

3

0 i

aij xj = 0,

aij = aji , i = 1, 2, 3,

j=1

/(a

ij )

1

$ P O Γ : S ≡ a x x = 0 QJ. Γ / (& Γ Q" O% Γ / Γ $Q" O/ 1/ / 2, / / 1 "& / " ,

Q" O$ 0 P , S = 0 / P ; P P . $Q" O/ P " & ( P ); P "& ( P +). 4.16 O/ (& [ O 35 P (p , p , p ) O Γ : S = 0 / Q(q , q , q ) Γ % P = Q. P Q (p + λq ) (i = 1, 2, 3). P Q Γ * λ 3

0

i,j=1

ij i j

p

0

0

0

0

0

0

1

1

2

2

0

3

3

i

i

Sqq λ2 + 2Spq λ + Spp = 0.

(4.20)

P / Æ a p = 0 (i = 1, 2, 3). S = 0, S = 0. Q ∈ Γ, # S = 0. ) λ *D (4.20) 5&Æ P Q Γ ( P (p , p , p ) Γ Q(q , q , q ) Γ (Q = P ), % P Q & Γ Æ (4.20) * λ S = S = S = 0. $ S = 0 3

j=1

ij j

pp

pq

qq

1

2

3

1

2

3

qq

pq

∂S ∂x1

p

q1 +

∂S ∂x2

p

q2 +

∂S ∂x3

p

q3 = 0.

pp

pq

0% 1%0),3

124

Q Γ 5

∂S ∂x1

= 0, p

∂S ∂x2

= 0, p

∂S ∂x3

= 0. p

$/ P O Γ / 0@

e$, $ O 1/ O"/ / & O2 O" " E O 3

Γ :S≡

aij xi xj = 0

(aij = aji ),

(4.1)

i,j=1

1 1. |a | = 0 Æ (a ) / 3 Γ Q" Γ A A A ' 2 A (p ), A (q ), A (r ).

E (p + q + r ) 6 $ §1.3 (1.10) 5 05 ij

ij

1

1

i

i

i

2

i

2

3

3

i

i









p1

x1

      ρ  x2  =  p2    x3 p3

q1 q2 q3

r1

x1



      r2 x ,  2  r3 x3

(1.10)

(aij = aji ),

(4.21)

5( (4.1) 5 3

aij xi xj = 0

i,j=1

A (1, 0, 0) ! A A 1

2

3

x1 = 0.

(4.22)

a11 x1 + a12 x2 + a13 x3 = 0.

(4.23)

+ A (1, 0, 0) 1

1%0))#"%. (4.22) (4.23) §4.4

(4.21) 5"

125

a12 = 0,

a13 = 0,

a11 = 0.

a23 = 0,

a22 = 0,

a33 = 0,

2 2 a11 x2 1 + a22 x2 + a33 x3 = 0.

(4.24)

%6

1 xi ρxi = |aii |

(4.24) "

(i = 1, 2, 3),

2 2 ±x2 1 ± x2 ± x3 = 0.

(4.25)

x , x , x 1

2

3

2 2 x2 1 + x2 − x3 = 0,

(4.26)

2 2 x2 1 + x2 + x3 = 0.

(4.26) 5 O ?* : O $ 2. |a | = 0 % (a ) / 2 Γ $/ Q" O / ' A (r ). A (q ), A ( A +) A A (p ). E (p + q + r ) 6 05$ Γ / 4.16, Γ Q"/ ) Γ Æ ' A 05 ( (4.1) " ij

ij

3

2

2

i

3

3

1

i

i

i

i

i

3

2 a11 x2 1 + a22 x2 = 0.

  1 x   ρxi = i |aii |    ρx = x , 3 3

(i = 1, 2),

(4.27)

0% 1%0),3

126

(4.27) "

2 ±x2 1 ± x2 = 0.

(4.28)

$ x , x (4.28) 5 1

2

2 x2 1 − x2 = 0, 2 x2 1 + x2 = 0.

x + x = 0 x − x = 0; Æ : x + ix = 0 x − ix = 0. 3. |a | = 0 % (a ) / 1 O Γ Q"%%$/ "

! x = 0, Γ " 1

1

ij

2

1

2

1

2

2

ij

1

3

aij xi xj = 0,

(aij = aji )

(4.29)

$ (0, p , p )(p + p = 0) / Æ (4.29), (4.29) a = 0, a = a = a = a = 0. Γ " i,j=1

2

3

11

2 2 12

2 3

13

x2 1 = 0.

22

33

(4.30)

- O 5 " ( > ):

     x21 + x22 − x23 = 0     |a | = 0, = 3  ij     x2 + x2 + x2 = 0   1 3  3  2     2 Γ : aij xi xj = 0  x1 − x22 = 0       (aij ) = 2 i,j=0     |aij | = 0  x21 + x22 = 0             (a ) = 1 x21 = 0 ij

O : O / Æ : /

5 O " /_34. O % "E "" O$E" " O$E /

1%0))#"%. 1 4.15 O

§4.4

Γ :

127

x21 + x22 + x23 + 2x2 x3 + 2x1 x3 − 6x1 x2 = 0

(∗)

"E 2 30/ (a ) = 3. # Γ Q" O # P (1, 0, 0) Γ ( &# =6 ), P Γ l : x − 3x + x = 0. Q(0, 1, 3) l Γ ($ = 6F&). Q Γ l : x + x = 0.

l × l = R(4, 1, −1) - Γ P QR. 5 P, Q, R ' A , A , A , % E (p +q +r ) = (5, 2, 2) 6$ §1.3 (1.10) 5 0 5      ij

p

1

2

3

p

q

p

2

3

q

1

1 x  1      ρ  x2  =  0    0 x3

2

3

i

i

i

x  1    1 1   x2  .   3 −1 x3 0

4

"E 05( (∗) 5" ρx

1

2 2 x2 1 + 16x2 − 16x3 = 0.

= x1 , ρx2 =

1 1 x , ρx3 = x3 , 4 2 4

( (∗∗)

2 2 x2 1 + x2 − x3 = 0.

E OQ" O

. /

1.

4.4

,#"-1-6()/ 1V0))..S*#"1Q..)

(1)

2x21 + x22 + 3x23 − 4x1 x2 − 4x1 x3 + 6x2 x3 = 0;

(2)

4x21 + 15x22 − 5x23 + 16x1 x2 − 8x1 x3 − 22x2 x3 = 0;

(3)

x1 x2 + x1 x3 + x2 x3 = 0;

(4)

x21 + 4x22 + 9x23 + 4x1 x2 + 6x1 x3 + 12x2 x3 = 0.

(∗∗)

0% 1%0),3 2. .' - 1/*,2R1V0) Γ ),N)CN*.)91),"+ Γ )..(-4

128

* c x x + c x = 0. 3. .',/SS1V0)70760*),"+//! - 19/(0))..(4* Γ : a x + a x + a x = 0 (a a a = 0) . Γ : b x + b x + b x = 0 (b b b = 0). 4. /31V0) 4x − 16x x + 16x − 2x x + 22x x + 7x = 0 ./* P (1, 0, 1), 1 1 3

1

2 2 2

2 1 1

2 2 2

2 3 3

2 1

1 2 3

1 2

2

2 2

1 3

2 1 1

2 2 2

2 3 3

1 2 3

2 3

2 3

Q(1, −1, 0), R(1, 1, −1). (1) (2)

"- P QR *741V0))!&RZ/*,) - P QR *A-1/*,(,741V0))..)

P&KP129:KJ + OQ" O ( '[ 4.12 O Γ 4.12 Γ ', Γ ,, Γ (. Γ (A, B, C, · · · ), ( (. Γ (a, b, c, · · · ), m Γ .

m Γ . ((! 4.13 Γ A, B, C, D, 2% MN, §4.5

(AB, CD) = S(AB, CD),

S Γ

! 4.24

1%* +)#"-6 $2 4.3, ,

§4.5

129

(AB, CD) = −1,

Γ " OPI. 4.14 x(P ) S Γ SP Γ P . ( S(P ) x(P ) & Γ (P ) EF. S(P ) ∧ Γ (P ) x(P ) ∧ Γ (P ). 2DN ( 2'% ($ %: WP 4.15 , S(P ) ∧ Γ (P ), S (P ) ∧ Γ (P ) % S(P ) ∧ S (P ). Γ (P ) Γ (P ), Γ (P ) ∧ Γ (P ). Γ (P ) ∧ Γ (P ) S, S Γ, Γ # "% $ 4.15, 4.17 4.17 (1) $ ! (2) $(& 1

1

1

1

! 4.25 > Steiner 4.15 4.17(1), ,

$! 4.26, Γ Γ A, A ; B, B ; C, C , P Γ P Γ P - AA Γ , Γ S, S . SB × S B = B , SC × S C = C . - B C . SP × B C = P . - S P Γ P , P !+ 4.26 B C Γ

0

0

0

0

0

0

0

0

0

0

0% 1%0),3 Γ EFG. P ) "- BB Γ Γ S, S , )+ (WP )?). "W$ P P Q 130

! 4.26

'[

4.15 " Γ = Γ , Γ (P ) ∧ Γ (P ) ' Γ [ . $P 4.18(Steiner) f Γ 2 p , f A, A ; B, B , P = AB × A B p 35 4.27, Γ A = B. A = f (A) B = f (B) + Γ C() C A, B), f (C) = C .

0

AB

0

PAB = AB × A B

PAC = AC × A C

% p = P P . ! Γ 8 AB CA BC , $ Pascal # P p % p 4.16 4.18 p Γ f G ! m G. 4 4.11 Γ 2 f ) p $ f ! $ 4.18 " Γ 2 f ) p 0

BC

AB 0

AC

0

0

0

0

1%* +)#"-6 131 Γ f Steiner $)Æ2 §4.5

4

! 4.27

Γ 2 f $) ! $ 4.18, " Γ 2 f ) p Γ X, X f 2 p Γ (*); Æ :> 4.17 Γ 2 f K) ! R L) ! MN) , f !!Æ : $$A$(*$ 4.27 4.28. 4.12

0

0

! 4.28 $ 4.14, 4.19 f : Γ (P ) ∧ Γ (P ) Γ (

0% 1%0),3 S(P ) ∧ Γ (P ), S(P ) ∧ Γ (P ), x(P ) ∧ Γ (P ), x(P ) ∧ Γ (P ). $ f ' ( S x f : S(P )∧S(P ), f : x(P )∧x(P ). % f $ $!A$!(*$ ⇐⇒f , f [$$!A$!(*$ 2 4.20 X, Y Γ 2 f P, P f (P = P ), (P P , XY ) = k(), k f ÆO+. 1 4.16 x &%Q" O Γ . x &% 2 4.29, x & A , A ; B , B ; C , C Q" O Γ . x (1) Γ S, S ∈ x. S " x A , A ; B , B ; C , C " Γ 2 A, A ; B, B ; C, C . (2) - AB , A B, P ; - BC , B C, P . (3) - P , P = p , p Γ X, Y . (4) S " X, Y " x X , Y . X , Y 132

1

S

S

1

x

1

1

x

1

1

1

1

1

1

1

1

1

1

1

AB

BC

AB

0

BC

0

1

0

1

1

1

! 4.29

x(A1 , B1 , C1 , X1 , Y1 , · · · ) ∧ S(A, B, C, X, Y, · · · ) ∧ Γ (A, B, C, X, Y, · · · ), x(A1 , B1 , C1 , X1 , Y1 , · · · ) ∧ S(A , B , C , X, Y, · · · ) ∧ Γ (A , B , C , X , Y , · · · ),

1

§4.5

1%* +)#"-6

133

Γ (A, B, C, X, Y, · · · ) ∧ Γ (A , B , C , X, Y, · · · ),

X , Y x " p Γ (:), x (*$ " p Γ S "4 x Æ p Γ $$ p Γ *A$ $ H Q " H'$$A$JK p Γ * T*+6 ,+R I p Γ !!+6

P>" !

'[S 4.18 Γ 2 f S, , Γ S f ( S f f S Γ # " ) $ ( 15( 2 DH 4.21 Γ , - (1) Γ $ ! (2) Γ 2 f ⇐⇒ f (3) Γ P , P ; P , P ; P , P " ⇐⇒ 1

1

0

0

0

0

0

0

0

1

1

2

2

3

3

(P1 P1 , P2 P3 ) = (P1 P1 , P2 P3 ).

4.19

Γ f ) p f SG. 0

! 4.30 $)#-

0% 1%0),3

134

Γ f , (1) f )$ ! (2) f $)! (3) f Γ *) $ 4.18, A$ !$$ !(*$) Γ ( ! Æ :) %- 4.23 Γ f Æ* −1. $ 4.22(3), Γ f ) p 4.24 Γ f Γ (( Γ Γ 35 0 - 4.31, P Γ P Γ A, A ; B, B ; C, C ; · · · . $ * P p . P = AB × A B, P = AC × A C, P = BC × B C & P p Γ 4.22

0

0

0

0

0

AB

0

AC

BC

0

Γ (A, B, C, · · · ) ∧ Γ (A , B , C , · · · ).

$ P = AB × A B p B, B #0@ AB

0

S

! 4.31

4.20

f

.

Γ f ) p Γ P 0

0

1%0))I#,3 4 4.13 Γ f $ !

§4.6

. /

135

4.5

Q!* T D 1V0) Γ )N) T X, T Y , X, Y *N*(4( T D Γ )R) T AB 0 Γ A, B. ,"+ (AB, XY ) = −1. 2. /+. (/ R * Γ +-!*( RX, RY %*0 AB P, Q. ,"+ (P Q, AB) = −1. 3. / A, B, C, D *1V0) Γ +0*(5 (AB, CD) = −1. ,"+!) AB . CD * 7 Γ )$E!)) 4. / A, A ; B, B *1%* Γ +/7)//.*( X, Y 2,&#-*),"+ A, B; A , B ; X, Y 2 Γ +2!&/7)///.*) 5. .! 4.32, / A 2,1V0) Γ . Γ )!&0*( P Q * Γ )K6(#/ Γ +! &+* P , AP, AQ %*0 Γ * P , Q . ,"+!) P Q 6(!&+*) 1.

0

! 4.32 ! 4.33 6. 74* x(P ) +/7 f )/76)/.*-C!,/SS1V0) Γ . , D f ) #-*) 7. .! 4.33, / A, B *#/1V0) Γ +)&+*( P P , P P *%*6( A, B ) Γ ),.K),"+ Γ (P ) . Γ (P ); Γ (P ) . Γ (P ) ?2 Γ +)/7* ). Γ (P ) . Γ (P ) 2G!+1*/7* . 8. / P P *1V0) Γ )(+* (#/ Γ +) ).K(M A, B * Γ +)&+*(5 Q = AP × BP , R = BP × AP . ,"+ Q, R /2!,1V0)+) 1

P&MM2S97Q + \= $"& "& % x = 0 "& # x = 0(l ) !

$ .2/*. §4.6

3

3

∞

0% 1%0),3

136

e &* O "& ! ! * ! Æ: K) ! RL) ! M N) , Q" K RL ! MN 4.21

Γ

Γ

(

l∞

),

Γ

Γ

,

.

! 4.34 ( O#$$A$ (*$" O Γ :

S≡

3

aij xi xj = 0

(aij = aji ),

(4.1)

i,j=1

x i

  S = 0,  x = 0, 3

Æ Γ l $5' ∞

a11 x21 + 2a12 x1 x2 + a22 x22 = 0.

(4.31)

x /x , 5 !! Æ : ⇐⇒     < 0   $$     a a =A = 0 ⇐⇒ Γ A$     a a    >0  (*$ $ 4.25 O Γ : S = 0, A F + O Γ (4.1), %Q" 1

2

11

12

12

22

33

33

1%0))I#,3 137

e 4.22 "& O Γ Γ . 4.26 $(* & A "&

§4.6

35

Γ C(c , c , c ). C l 1

(

3



 a11

   a12  a13

Æ

2

a12 a22 a23



: x3 = 0





c1

a13

∞

0

        a23   c2  = ρ  0  ,     1 a33 c3

ρ = 0,

(4.32)

  a11 c1 + a12 c2 + a13 c3 = 0,  a c + a c + a c = 0, 12 1 22 2 23 3

(4.33)

 ∂S = 0,   ∂x 1 C   ∂S = 0, ∂x

c : c : c = A : A : A . # $(* : (A , A , A ), A : A$ "& c

(4.34)

2 C

1

2

3

31

32

31

33

32

(A31 , A32 , 0).

33

  a11 c1 + a12 c2 = 0,  a c + a c = 0, 12 1 22 2

3

= 0,

(4.35)

Æ (a , −a , 0) ! (a , −a , 0). $ 4.22 -% C O Γ ( & C Γ . $"& A !\ $(* $ e, A e. 2 O" 2 A , $

n C?n 4.23 "& O Γ & Γ n. O Γ "& C?n. Γ Æ Γ C?31. 12

11

22

12

33

0% 1%0),3

138

$ 4.23 Æ- 4.27 O Æ (V% Æ Γ *A % ( Γ & * V $ 4.27 A A a x + a x + bx = 0, ! a x + a x + bx = 0 (b ). (4.36) 7 O Æ O Γ "& (1, 0, 0), (0, 1, 0), () 11 1

12 2

3

12 1

22 2

3

∂S = 0, ∂x1

Æ

l1 : a11 x1 + a12 x2 + a13 x3 = 0, l2 : a12 x1 + a22 x2 + a23 x3 = 0,

∂S = 0. ∂x2

Γ l ∂S ∂S +k = 0, ∂x1 ∂x2

l:

k ∈ R.

(4.37)

5 k l Æ 1'$5 l (1, k, 0) Æ & (4.37) l Æ l , l l "& (a + ka , −(a + ka ), 0) l ∂S ∂S ∂S ∂S (a + ka ) − (a + ka ) = 0, Æ +k = 0. (4.38) ∂x ∂x ∂x ∂x

12

1

12

22

22

2

Æ

11

11

12

12

k = −

1

2

a11 + ka12 . a12 + ka22

a22 kk + a12 (k + k ) + a11 = 0,

a11 a22 − a212 = A33 = 0.

(4.39)

(4.40)

5Æ l, l Æ (& O ( ( Æ 1 4.17 O Γ : x − y + 3x + y − 2 = 0 V l : 2x + y = 0. 2 (4.40)

2

2

∂S ∂S +λ = 0, ∂x ∂y

§4.6

1%0))I#,3

139

Æ x + 32 + λ −y + 21 = 0 λ V1 ' −2. λ = −2. #

OT

1

2x + 4y + 1 = 0.

4.35 T O Γ V QQ O V Γ U, U .

4.18

QQ

! 4.35 0 (1)

35

! 4.36

QV = V Q ; (2) OU 2 = OV · OT.

% QQ "& X . O Æ O l T QQ X $ X T . X O X OT. $

(1)

∞

∞

∞

∞

∞

∞

(QQ , V X∞ ) = −1,

V QQ Æ V Q = QV. (2) T QQ (U U , T V ) = −1. 5 O U U I

OU 2 = OV · OT.

1 0 O V , 35 4.19

4.36).

P1

P1 (y1 , y2 , y3 )

(1)

HK

HK

P ( 1

HK

l∞

Q∞

(HK, P1 Q∞ ) = −1.

# P , Q O Γ Æ Q P Æ HK P Q HK P 1

1

∞

∞

∞

1

1

0% 1%0),3 (2) " P (y , y , y ) P Γ S = 0. l = 0 # HK

140

1

x3

1

2

3

1

y

∞

Sy + λx3 = 0,

Q (( P (y , y , y ) HK S S ∞

1

1

λ=−

HK

Sy −

2

yy

y3

3

=0

,

Syy x3 = 0. y3

eop O"&

& * O op

]3 O op31 4.24

.

yy + λy3

.

$$]3(*:]3A] 32+ O ]3 Æ $ Æ ! O ((]3 O] 3 Æ 1 4.20 O Γ :

S≡

3

aij xi xj = 0.

(aij = aji ), |aij | = 0, A33 = 0

(4.41)

i,j=1

Γ ]3

]3 2h &% a22 kk + a12 (k + k ) + a11 = 0.

%k=k,

a22 k 2 + 2a12 k + a11 = 0.

Æ&%0Æ]3" A < 0, k , Γ $" A > 0, Æ : k , Γ (* k ( 33

i

33

i

∂S ∂S +k = 0, ∂x1 ∂x2

i

1%0))I#,3 Æ]3 ∂x∂S = 0 ! ∂x∂S = 0 ]3 \ 2h ]3 §4.6

1

141

2

  S = 0,  x = 0, 3

a11 x21 + 2a12 x1 x2 + a22 x22 = 0.

"& Æ Γ "& ]3" a11 x2 + 2a12 xy + a22 y 2 = 0.

(ξ, η). ]3

2h

a11 (x − ξ)2 + 2a12 (x − ξ)(y − η) + a22 (y − η)2 = 0.

* Γ P Γ * S S = S . " P P (A , A , A ), (Æ]3 " pp

31

32

33

∂S ∂S x1 + x2 + ∂x1 p ∂x2 p ∂S = 0, ∂S = 0. ∂x1 p ∂x2 p ∂S Sp = x3 . ∂x3 p

Sp =

P P (

2 p

∂S ∂x3

p

x3 ,

Sp = (a31 A31 + a32 A32 + a33 A33 ) · x3 = |aij |x3 .

]3

Spp = |aij |A33 ,

Æ , ]3

|aij |A33 S = |aij |2 x23 ,

% λ = −|a

ij |/A33

S + λx23 = 0.

A33 · S = |aij | · x23 .

(4.42)

142

1 2h

4.21

$ x + 3xy − 4y ]3 2

2

0% 1%0),3 + 2x − 10y = 0 ]3

∂S ∂S +λ = 0, ∂x ∂y

λ λ

1

1 = − , λ2 = 1. 4

1 + 3λ − 4λ2 = 0,

( (∗) ]3

5x + 20y + 18 = 0,

2h

(∗)

5x − 5y − 8 = 0.

(−14, 26, −25) ]3

2h

x−

14 25

2

14 26 ,− . 25 25

2 14 26 26 +3 x− = 0. y+ −4 y+ 25 25 25

$ (4.42) 1 1 3/2 3/2 −4 −5 1 −5 0 + λ

, ( S + λx λ = − 144 25

2 3

=0

=0

"]3

25x21 + 75x1 x2 − 100x22 + 50x1 x3 − 250x2 x3 − 144x23 = 0.

1 $*]30 * 35 *]3 $ 4.22

4.37,

M

P, Q, C

P Q × l = N , - CM, CN , CM × l = M . C l M P Q, CM l × P Q = N . CN M CM, CN Æ ∞

∞

∞

∞

∞

∞

∞

∞

∞, ∞.

∞

C(P Q, M N∞ ) = (P Q, M N∞ ) = −1,

# M P Q 1 4.23 0 $*]3L(

1%0))I#,3 143 35 4.38, $*2]3 A, B A , B . *]3"$ * 4.10 ) ( *Q" 6$'$!* ) AB , A B, l Æ AB //A B. ( A B A AB B ( ABC A B C §4.6

∞

! 4.37

! 4.38

. /

1.

4.6

9Z-/ 1V0)2N9.?)0)(-,+-0)) ( - 1)

(1)

2x21 + x1 x2 + x22 − 6x1 x3 − 5x2 x3 + x23 = 0;

(2)

x21 − 2x1 x2 + x22 − 2x1 x3 + x2 x3 − x23 = 0.

,1V0) Γ : x + 2x x + 2x + 4x x + 2x x + x = 0 )(* A(1, 1, 1) )!J C,$E!J) 3. E+1V0) Γ : x + 4x x − 2x + 10x x + 4x x = 0. (1) "- Γ 2^0)/ (2) , ( - 1/ (3) ,8-* 3 2 )!JC,$ E !J/ (4) ,^_)..) 4. ,/ ^0))^_)..) 2 1

2.

2 2

1 2

2 1

1 2

(1)

x2 + 2xy − 3y 2 + 2x − 4y = 0;

(2)

xy + y 2 − x − 3y − 2 = 0;

(3)

xy − a2 = 0.

1 3

2 2

2 3

1 3

2 3

2 3

,1V0) Γ : x + xy + y = 1 . Γ : 3x − xy + 2y = 1 )@$$E!J) 6. ,"+T0)-N))N*.)2!,!J) 7. ,"+(#/^_)+)!+*4 D 1V0)SK *)HI912!,1V0)) 8. .! 4.39, R635* P , Q RT0)N)(91!&EN0,, ABCD. ,"+; &EN0,,)/*) AC . BD )0* O 2T0) () 5.

1

2

2

∞

∞

2

2

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Q^0)+-!*D-^_))!)),"+;!).^_)41-0,, )R*+# (F`! 4.41). 11. .! 4.42, / P P 23(1V0))!J)-N* Q O)N). P O)N)0 R, P Q × P R = X. ,"+ P R = RX. 12. .! 4.43, / T *BQ))K P Q )Z*(( T )!J0K P Q N . "-+)M T N RBQ)%) 10.

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145

P&MM2S9QG + Æ #" "E$*:6"& x = 0 # x = 0 !Æ ' A , A "& O §4.7

3

3

1

2

Γ : S≡

3

aij xi xj = 0,

(aij = aji )

(4.1)

i,j=1

5E 0[g e$, |a | = 0, / (a ) = 3, 2 A ,$ 1. A = 0. Γ OÆ(*!$"& *#

A , Æ l A , A . Γ 6# 6" §4.5, Γ " ij

ij

33

33

3

1

∞

2 2 ±x2 1 ± x2 ± x3 = 0.

2

(4.25)

$ x , x x 7# (4.25) " ( > ) ).    x +x −x =0 (* A >0 (4.41)   x +x +x =0 :(* A <0 x −x −x =0 $ (4.42) 2. A = 0. Γ " OÆA l !

("& *) A , Γ & A , A * l A , 6# 6" Γ " 1

2

3

33

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2 1

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2 3

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33

∞

1

3

3

2

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3

A (0, 1, 0) a (4.43) a 2

aij xi xj = 0.

i,j=1

21 x1 + a22 x2 + 21 = a23 = 0,

(4.43)

5 x 0. a =

a23 x3 = 0,

a22

2 = 0 11 = a12

= 0,

0% 1%0),3

146

a13 = 0; a32 = a33 = 0, a31 = 0.

(4.43) Æ

a22 x2 2 + 2a13 x1 x3 = 0.

(4.44)

%6

 a   ρx1 = 22 x ,   a13 1   ρx2 = x2 ,       ρx = x , 3 3

( $> ) )

x22 + 2x1 x3 = 0.

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AE 4.5 2 -" -Q" O"

[g e$, |a | = 0, 1 ≤ /(a ) ≤ 2. 1. / (a ) = 2. Γ Q"%$/ (1) / & "& A , A , " §4.5, Γ " ij

ij

ij

1

2

2 ±x2 1 ± x2 = 0.

(4.46)

x , x (4.46) Æ (> ) x − x = 0, ; (4.47) Æ :. x + x = 0, (2) / "& %"& / A , +"& A , " §4.5, Γ " (> ) x − x = 0, ; (4.48) :. x + x = 0, (3) / "& %"& "& I" Γ . 2 / A , "& A , 1

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1%0))I#%. 147 & A , 6# 6 O " x x = 0, %"& . (4.49) 2. / (a ) = 1. $ §4.5 Γ Q"% $/ " (1) / & x = 0, Γ " x = 0, %. (4.50) (2) / "& x = 0, Γ " x = 0, "& . (4.51) -Q" OL" 1 4.24 " O Γ E §4.7

3

1 3

ij

1

2 1

3

2 3

2

x21 + 2x1 x2 + 2x22 − 6x1 x3 − 2x2 x3 + 9x23 = 0.

1 −3 1 |aij | = 1 2 −1 = −4 = 0, −3 −1 9

Γ Q" O A33

1 = 1

1 = 1 > 0, 2

# Γ (* # A = 5, A = −2 (5, −2, 1). R(5, −2, 1) ' A . l Γ P (1, 0, 0) A . P l Q(1, −1, 0) A . A A A Γ

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32

3

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∞

2

1

2

1 3

0% 1%0),3 $ 6# (1.10) 5 05      148

x  1    −2   x2  ,   1 x3

1 1 x  1      ρ  x2  =  0 −1    0 0 x3

5

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2 2 x2 1 + x2 − 4x3 = 0.

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x1

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1 0

      µ  x2  =  0 1    x3 0 0

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(1)

x21 + 4x1 x2 − 2x22 + 10x1 x3 + 4x2 x3 = 0;

(2)

x21 − 2x1 x2 + 2x22 − 4x2 x3 + 3x23 = 0;

(3)

4x21 + 4x1 x2 + x22 + 4x1 x3 + 2x2 x3 − 48x23 = 0;

(4)

2x21 − 2x1 x2 − x22 + 2x1 x3 + 5x2 x3 − 8x23 = 0;

(5)

x21 − 2x1 x2 + x22 − 2x1 x3 + 2x2 x3 + x23 = 0.

2.

x1

2 2 x2 1 + x2 − x3 = 0.

(* 1.

 0

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2

(αx + βy + γ) + 2(px + qy + r) = 0

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149

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2 Heath +*[2 Euclid L §5.1 Euclid

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UU Desargues S *_$ Pascal Desargues Pascal h h $ $U"%2 Euclid -,G( Euclid c $ "& &% Euclid $[d O Desargues Pascal iQ( ie 19 WQ Gaspard Monge(1746∼1818) Lazare N. M. Carnot(1753∼ 1823) 3W-S Jean Vicotor Poncelet(1788∼1867) ". 3W Poncelet @Vd 1812 \e *b `T j`X jV`\b 1813 jV`hihii e& Poncelet 4g f]!kk<^l Æ+ ;_d ,_ 1814 6 d% 7 i F!4Q4Q `Æklj I, 1822 Vd Poncelet #f( 1862 J 1864 L 17 WQ 154

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Æ% /NYY Euclid " G= Augustus Ferdinand M¨obius(1790∼1868) Julius Pl¨ ucker(1801∼1868). Pl¨ucker U 0(2 Pl¨ ucker mc Felix Klein(1849∼1925). Poncelet GP ! 3% Edmond Laguerre(1834∼1886) Arthur Cayley(1821 ∼1895), Euclid kk Euclid Th h $W$GP e 1 Klein %#jA 5 Euclid $(* Euclid ) GP7 Euclid % Euclid Euclid ! 1872 Klein C( Erlangen 3E % ' *#+E 2 E Erlangen KH Klein KH% Euclid Euclid % G(7 KH ?&WQS 19 WQ) * Klein Erlangen KH; " 3V*J Klein 156

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+ 3% 7 l ^ $ 1619 11 d 10 hbT!jjcm R X Descartes ,- # LL%* 1 L Lbfc L 1 kd a, b a ab ? D !_ =:* 2$ Æ 4 Pappus 1 (_ x y h F& & LO *> *_ Descartes i 1)Fermat gC[n]hJijG x, y, z Ao n > 2 ko x + y = z . dC[n 1994 eo Andrew Wiles i R. L. Taylor eAk^R?p A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, §5.3 Descartes

1)

2

n

n

n

Ann. Math. 141(3)(1995), 443−551; R. L. Taylor and A. Wiles, Ring-theoretic properties, Ann.

Math. 141(3)(1995), 553−572.

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U]' 3% van Schooten(1615∼1660), John Wallis (1616∼1703), Isaac Newton(1642∼1727), James Bernoulli(1655∼1705), La Hire (1640∼1718) TY Newton James Bernoulli 7( 1729 Jacob Hermann(1678∼1733) 5eW 1)Gottfried Wilhelm Leibniz(1646∼1716) ADGdkjm;ge&DGm;ikrm; 158

1)

Æ@/$k./4/N 159 Df $$( 5 18 WQ 43n f* Fermat, Descartes La Hire "g 18 WQ Antoine §5.4

Parent(1666∼1716), John Bernoulli (1667∼1748), Alexis-Claude Clairaut(1713∼1765)

Jacob Hermann, Leonhard Euler (1707∼1787) TY (%

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% 1

3% Euclid G?;G?J"U]E "1Æ ' % , Æ ,$ 9 25; ,25 " 9 25 ?U]"=%ifXJ1 l 2 ieUf = $ c (S;5. Euclid hÆmlW Nikolai Ivanovich Lobatchevsky(1793∼1856) .j V` Novgorod je2fo 7 ngj:ok sm 15 n (bQ Kazan – 19 nnf< Kazan 22 n 24 n 1820 1825 l Kazan – 1827 1848 l Kazan ? # G¨ottingen m%i*@ TYY Kazan Wh Lobatchevsky klgNQ Y 1893 Kazan d Lobatchevsky Q Lobatchevsky 0 0 +h u + qEL25 §5.4

Æ% /NYY ( (25 <) "GP#4

Euclid +

: (( 180 2 180 XJ(()!( 360 ?2 360 >(( ( X 2=](!](+! · · · · · · mj:mj 1826 2 d 23 Lobatchevsky Kazan – *e U 0 [ h Kazan Uj-,on V ` \ Lobatchevsky 2L+ 1829∼1830 Kazan i 1835∼1837 iU$ 1837 U: 1840 RU Q * Lobatchevsky S Lobatchevsky *dD 2000 D kappn Euclid qm d l b* !j2[nXJnrnojQ eU eW)a )qm)oq)or'a "o $ Lobatchevsky psmq[ .)!qW $\ UjU= Euclid 4 Lobatchevsky ) W3 20 W L) Lobatchevsky Euclid Carl Friedrich Gauss(1777∼1855) pl John Bolyai(1802∼1860). Gauss 19 WQ _0 Euclid ) 1816 BN Euclid 5" Gauss )t X ,K Beotians ssÆK Gp=oo Gauss ;+XJp o S$m Yc"((30 Euclid Euclid $S-,) 1 Gauss , W _ #[ Gauss 1846 11 d 28 Zh Schumacher [ ) W Lobatchevsky U · · · · · · Lobatchevsky b c 160

.t%/N 161 k# $ 54 *

Lobatchevsky ,

. ^Z 1Q 1 k $ k Bolyai plb_jo Gauss Zh Bolyai t n 0; Euclid 1823 11 d 23 jo[)

q + /W!jo Bolyai ,[NY Gauss Gauss qR4[) 40 n-,l Bolyai 4[Q Gauss v u, !n 1 Bolyai Euclid A UA j q_ 1832∼1833 Bolyai Lobatchevsky Bolyai Lobatchevsky *^r .U Gauss Bolyai U Bolyai jo[ )g*mU,g . , Gauss r Bolyai !n Euclid u

bq Riemann Euclid $ 1 Eugenio Beltrami(1835∼1900), Jules Henri Poincar´e (1854∼1912), Felix Klein TY Euclid ! DfL Euclid Lobatchevsky :m ^Z Euclid A Lobatchevsky $ Riemann (* )%%' !5 Klein %% Euclid 19 WQ3Zm* E# LO E# W! e 1 1 20 W Q2 §5.5 Gauss, Riemann

QuQNPK L Euclid 17 WQ@r Newton, Leibniz = tar % v7 . n/ §5.5

Gauss, Riemann

Æ% /NYY i $ *F$ Luigi Bianchi(1856∼1928) 1894 ?X .1 17 WQ ' 1 ',3(S 1X e)(21 XK h os O I 18 WQh n 00- Christian Huygens(1629∼1695) `o ] ]v' '& Newton ' '&=** John Bernoulli 3(-, Alexis-Claude Clairaut 16 n$' 17 n(#Q $' &= " ' ' A?5 5 Euler A? ** Clairaut ' E"+' w '$ Michel-Ange Lancret (1774∼1807) 5 *w'ÆA ?"'Cauchy, Alfred Serret(1819∼1885), Fr´edederic-Jean Fr´enet(1816∼1900) VA?'w' Serret-Fr´enet 5 SJohn Bernoulli, James Bernoulli, Euler TY 1760 Euler Gaspard Monge(1746∼1818) 71 TY f Euler, Monge1807 Monge ^!1 & a) rn5Z3 Monge L Desargues - Monge + FH+ TY 162

.t%/N 163 "pci TY Monge Y (pH5O J 19 WQu [VdÆY Monge k (Ecole Normale). Monge %E( , F $ 3Yc Carl Friedrich Gauss(1777∼1855), W $as 1 100 #!)D vTY) W!! Gauss <s Gauss [ $ c%Y %[U S H+qTY Gauss 1816 SX UYWP 1827 U2 3T Euler * $>6** $** Gauss 20A?6('S " .A?! , . $J+6"S Euclid A?5 *@# # A?5 Gauss 2$ Georg Friedrich Bernhard Riemann(1826∼1866) LL Riemann .RQCjo o 6 n2 14 n( w 19 n( G¨ottingen Gauss 5 sfsfU ( Riemann ! 3@ 1859 U 2% !rU Riemann t Hilbert 23 1 8 1 G¨ottingen j s Q!E ttUx t % 39 n QS 2=) *Fr Riemann t Riemann Riemann 1 Riemann Riemann # Riemann Riemann Riemann ζ @ TY@ 1854 Riemann G¨ottingen a $ Gauss §5.5 Gauss, Riemann

Æ% /NYY EE 1868 1861 Riemann oVd scmD5U VdUE eW 19 WQ Riemann h Riemann %3% Euclid Euclid %I E#$ Gauss, Lobatchevsky, Bolyai

$1 [

6 Riemann & I# Euclid ) )32H

"Q3^Z5 - Riemann Gauss Euclid CC Riemann n n ) n *(D n ) n *) *+" fI))')'$ $) .!" ) () % #+ Euclid 2 p V 2') ')) $') Euclid D' (Æ a)). Euclid 3> " Lobatchevsky $[[U2JF , 20 WQ Albert Einstein(1879∼1955) u ' qQ# Zh Marcel Grossmann(1878∼1936), ) Grossman, q , o Grossman NY Bernhard Riemann w , 60 %p Y Einstein o5 E = mc , Y Riemann qW 1868 UY!pWP q ( 2 Riemann LL Eugenio Beltrami(1835∼1900), Elwin Bruno Christoﬀel(1829∼1900) Rudolph Lipschitz (1832∼1903) 55 E( Riemann LL5Gregorio RicciCurbastro (1853∼1925) Tullio Levi-Civita(1873∼1941) ?5, 5 1901 m-UA, Einstein !m!!m0! 164

2

Q Cantor . Poincar´e ('r 165 " 3Z2!52! S . p ?S?# " ? !! Riemann ' !5 Einstein – Riemann ' n = 4 !5.u! Riemann WP? 1917 Levi-Civita D ( Riemann ' ttT Elie Cartan(1869∼1951), ( $ !)2 O §5.6

a Cantor D Poincar´e `sK U topology Z@Y 19 WQ 20 WQ 7 eP>)w ( .<) , ( Tr) :6%='!" ()"$ 0+ $ tt u 1872 Klein Erlangen KH H+U+ 20 WQ e Cantor 0 ' L !^ h eLQi e Georg Cantor (1845∼1918) jQ

j:eaRQ 1863 ( Z 1867 sf 1869 l Halle 1879 l ÆÆ 1874 Cantor 29 nZs (Journal f¨ur Mathematik) "&m)Ush Cantor L+fU 1897 19 WQ& ="K _CE" qv"& O"& 3wtt! s !!E! L(XJGaY*! Cantor yu asÆSH+x \ ""& (0%% §5.6

Æ% /NYY % Æ %rW? ÆaP q)Æ$ l"& ℵ ( ℵ Z`=:), 2"& c (+

Continuum =:), 0 c > ℵ . 0$ ℵ 2 , % c= 2 . ℵ L ℵ , % ℵ ≤c. ℵ =c ,"0! + 1900 W! Hilbert 23 1 Cantor 'W'1 uv

# e b [n Pmof ℵ s sX+ Leopold Kronecker (1823∼1891) hPn[ n 10 W Cantor ; $, aostr e6 Cantor, q' Hilbert, Cantor RQ Dvg Cantor f* R5Z c Cantor Sz [ Bertrand Russell(1872∼1970) Cantor " tsY ! Cantor % %S J -$ (@ Hilbert Banach L 4@os U4&$ -" $ Euclid -"2, -"+6Q Maurice Fr´echet (1878∼1973)1906 sfU 4@ E((4@@ 4&> $4@ Fr´echet -"u{ Euclid @4& @2 (XJ L+$L+ Felix Hausdorﬀ(1868∼1942) 1914 K L+4&I)! 2 Fr´echet, Hausdorﬀ Paul S. Urysohn (1898∼1924), Poincar´e, Luitzen E. J. Brouwer(1881∼1967) U2o 166

0

0

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1

1

0

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1)

W. Haken, Every map is four-colorable, I, Discharging, Illinois J. Math. 21(3)(1977), 429−490; K.

Appel and W. Haken, Every map is four-colorable, II, Reducibility, Illinois J. Math. 21(3)(1977), 491−567.

Æ% /NYY

), 1894 1904 10 6 m UQYUY2 Betti (Eurico Betti(1823∼1892), ), 65 2 Euler 5 Euler-Poincar´e ) eN —— Poincar´e U 3 Poincar´e !X + —— X Poincar´e t < '2' Poincar´e 0 ) , + B *

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Hilbert

.u/NEzv 169 !A Q Ug+[ Bertrand Russell 1902 )

Euclid & #$|?h}!kd 5b[n'Y{Y{$ GPlf %22f $"q0 0 !0Æ+ 2:60 j3 Euclid u0 |xw · · · · · · . 19 WQ 70 7 3 $6Z 5" D Moritz Pasch(1843∼1930), 1882 1926 Euclid = "% $ Giuseppe Peano(1858∼ 1932) + TY e4 Hilbert David Hilbert(1862∼1943) ^Dyf K¨ onigsberg , % Euler '/ iLx1 Yw= ! # K¨onigsberg. Hilbert 2q

[ p WP 1880 ( G¨ottingen # 2nw Hermann Minkowski(1864∼1909) Adolf Hurwitz(1859∼1919) Hilbert : [xE 5 z!y,bU) ,]#! !] -;.hx 0 π f}W Ferdinand Lindemann(1852∼1939) Hilbert sfUuc Paul Gordan(1837∼1912) x\ -, 1885 2 d 7 sf L+ H+; 1888 %!R/% Gordan zyvJ) Hilbert yd 1 sf RQQÆ1 3% Klein, Poincar´e 1892 G¨ ottingen q Hurwitz l 1895 l G¨ ottingen fw 1930 Qc G¨ottingen !y 1) nxzzxy4{zy §5.7 Hilbert

1)

Æ% /NYY

170

W! Hilbert 20 WQH`" qTY 1900 zW! Vd2 Hilbert 8 d 8 ziN 23 1 Hilbert 1 w iNh 20 WQ K HU& Minkowski Hilbert [ ) W! "Q qE% E [qt —— , $y Hilbert W!WY "{c G¨ ottingen Hilbert |x x # $$ # # Q Hilbert h 100 Richard Courand(1888∼1972) E ) Hilbert y$ 0 · · · · · · 5Z |{ ,L- ![ Hilbert *~SÆ'[ :6 ) Hilbert a5L:T= Euclid 6 3 Euclid %e_c Hilbert Hilbert 1899 L 1898 {[ G¨ottingen y) Hilbert R{0+

1891 *) *) y y}zzq Hilbert " !& %yy}zz!

^Zq · · · · · · · · · · · · (

( Gauss, Riemann, Klein

&y?

I.

" . A B, a " . A B, " . JJ " . A, B C, ! α, ! J " . A, B C, ! 3% " . , α *

α " . ,! α β A, * J + B 1 2 3 4

5

6

7

§5.7 Hilbert

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171

" . ! J 8

II.

`a?

# . , B A C * A, B C % B C A # . A C, AC J B, C A B # . A B a A, B ! B, A AB. A B ! A B " a ) Æ # . (Pasch ) A, B C a A, B, C ()A, B ! C() , a AB * AC ! BC 1

2

3

4

III.

SU?

$ . , A, B a A a !+ a * a A _ (w ), , B , AB A B AB ≡ A B . $ . , A B A B AB * A B ≡ A B . $ . AB BC a A B B C a , AB ≡ A B , BC ≡ B C , * AC ≡ A C . $ . ∠(h, k) α a α a α ! _ h a O α % k , ∠(h, k) ∠(h , k ) % ∠(h , k ) a _ ( . $ . ( ABC A B C , AB ≡ A B , AC ≡ A C , ∠BAC ≡ ∠B A C , * ∠ABC ≡ ∠A B C .

1

2

3

IV.

b?

5

4

a A a * a A A a V.

zc'?

{ . (Archimedes ) , AB CD * AB "+ A , A , · · · , A , AA , A A , A A , · · · , A A CD B A A 1

1

2

n

n

1

1

2

2

3

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Æ% /NYY { . (E) " I , I , II, III V % f L+

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2#4 7 , ( ` J( 0 E$ 2 Poincar´e Hilbert ÆX ) l L, , Hilbert Æ *

l! Æ { |~ {&|{ "7 Hilbert Hilbert a, b, c 172

2

I.

1

2

1

z{?

" . a b ! c, F a + b = c ! c = a + b. " . " a b x y, a + x = b y + a = b. " . ! 0, a a + 0 = a 0 + a = a. " . a b +J ! c, F 1

2

3

4

! c = ab. " . " a b % a 0, x y, ax = b ya = b. ab = c

5

.u/NEzv " . ! 1, a a · 1 = a 1 · a = a.

§5.7 Hilbert

173

6

II.

BC?

# . a + (b + c) = (a + b) + c. # . a + b = b + a. # . a(bc) = (ab)c. # . a(b + c) = ab + ac. # . (a + b)c = ac + bc. # . ab = ba. 1 2 3 4 5 6

III.

`a?

$ . " a b + 2 F a > b b < a. $ . " a > b % b > c, a > c. $ . " a > b, a + c > b + c c + a > c + b. $ . " a > b, c > 0, ac > bc ca > cb. 1

2 3

4

IV.

zc?

% . (Archimedes ) " a > 0 b > 0 a

. 1

a + a + · · · + a > b.

% . (E) (&% ( y)"& :6Æ f Hilbert 7 I *"|xw Hilbert l! 0 Hilbert Y!"e e 2" '1 1 H@ Karl Weierstrass(1815∼1897) @ + Q * 2

Æ% /NYY Jordan 1890 Peano F Jordan QWI '+ Peano Hilbert 6 +# Helge von Koch(1870∼1924) 1906 +%Lx ?Q"& %1 3Z" x\ 20 WQ

h 1 CyF ]Z "!"]Z|s"&" t"+j 174

|}de [1] Frank Ayres, Jr. Schaum’s Outlines of Theory and Problems of Projective Ge-

|z~)C|}})z}+~}}}+~z( 1990 .~:(~)#"/N)}+}0}}+~z( 1983 M. Kline. ~=C{@)}C9C||2|)+++|~}+~z( ometry. McGraw-Hill Book Company, 1968

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1979

}~|)C~ —— }|~{~4)}+|+~z( 1998 [6] ~5-(|(|5})}0/N)}+}0}}+~z( 1983 [7] ~5-(|(|5}({)}0/ND.:)}+}0}}+~z( 1994 [8] }~)~|%||T)}+|+~z( 1992 [9] ~;(@~.)C)}+}+~z( 2000 [10] ~;(@~.)CR<)}+}+~z( 2000 [11] ~:)}0/N)}+}0}}+~z( 1983 [12] ~)}0/N)}+}0}}+~z( 1983

[5]

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