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O
0
2
(b
1
2
0
: b
2
2
0
: b
3
2
0
) O
0
3
(b
1
3
0
: b
2
3
0
: b
3
3
0
) E
0
(c
1
: c
2
: c
3
)
x
1
= b
1
1
0
ρ
1
0
x
1
0
+ b
1
2
0
ρ
2
0
x
2
0
+ b
1
3
0
ρ
3
0
x
3
0
x
2
= b
2
1
0
ρ
1
0
x
1
0
+ b
2
2
0
ρ
2
0
x
2
0
+ b
2
3
0
ρ
3
0
x
3
0
x
3
= b
3
1
0
ρ
1
0
x
1
0
+ b
3
2
0
ρ
2
0
x
2
0
+ b
3
3
0
ρ
3
0
x
3
0
,
ρ
1
0
ρ
2
0
ρ
3
0
b
1
1
0
ρ
1
0
+ b
1
2
0
ρ
2
0
+ b
1
3
0
ρ
3
0
= c
1
b
2
1
0
ρ
1
0
+ b
2
2
0
ρ
2
0
+ b
2
3
0
ρ
3
0
= c
2
b
3
1
0
ρ
1
0
+ b
3
2
0
ρ
2
0
+ b
3
3
0
ρ
3
0
= c
3
.
x
1
0
= a
1
0
1
x
1
+ a
1
0
2
x
2
+ a
1
0
3
x
3
x
2
0
= a
2
0
1
x
1
+ a
2
0
2
x
2
+ a
2
0
3
x
3
x
3
0
= a
3
0
1
x
1
+ a
3
0
2
x
2
+ a
3
0
3
x
3
,
¯
¯
¯
¯
¯
¯
¯
¯
a
1
0
1
a
1
0
2
a
1
0
3
a
2
0
1
a
2
0
2
a
2
0
3
a
3
0
1
a
3
0
2
a
3
0
3
¯
¯
¯
¯
¯
¯
¯
¯
6= 0 .
x
0
=
a
1
0
1
x + a
1
0
2
y + a
1
0
3
a
3
0
1
x + a
3
0
2
y + a
3
0
3
, y
0
=
a
2
0
1
x + a
2
0
2
y + a
2
0
3
a
3
0
1
x + a
3
0
2
y + a
3
0
3
.
A(a
1
: a
2
: a
3
) 7→ A
0
(a
1
0
: a
2
0
: a
3
0
) ,
B(b
1
: b
2
: b
3
) 7→ B
0
(b
1
0
: b
2
0
: b
3
0
) ,
C(c
1
: c
2
: c
3
) 7→ C
0
(c
1
0
: c
2
0
: c
3
0
) ,
O20 (b120 : b220 : b320 ), O30 (b130 : b230 : b330 ), E 0 (c1 : c2 : c3 ), îïðåäåëÿåòñÿ ñîîòíî-
øåíèÿìè:
x 1
= b 1 10 10
0ρ x + b 1 20 20
0ρ x + b 1 30 30
30 ρ x
1 2
0 0 0 0 0 0
x2 = b210 ρ1 x1 + b220 ρ2 x2 + b230 ρ3 x3
x3 = b3 ρ10 x10 + b3 ρ20 x20 + b3 ρ30 x30 ,
10 20 30
0 0 0
ãäå ρ1 , ρ2 , ρ3 îïðåäåëÿþòñÿ èç ñèñòåìû óðàâíåíèé:
1 10 1 20 1 30 1
b10 ρ + b20 ρ + b30 ρ = c
0 0 0
b210 ρ1 + b220 ρ2 + b230 ρ3 = c2
b3 ρ10 + b3 ρ20 + b3 ρ30 = c3 .
10 20 30
Ïðîåêòèâíûì ïðåîáðàçîâàíèåì ìíîæåñòâà òî÷åê ïðîåêòèâíîé ïëîñ-
êîñòè íàçûâàåòñÿ òàêîå âçàèìíî-îäíîçíà÷íîå ïðåîáðàçîâàíèå, ïðè êî-
òîðîì ëþáûå òðè òî÷êè, ëåæàùèå íà îäíîé ïðÿìîé, ïåðåõîäÿò â òðè
òî÷êè, òàêæå ëåæàùèå íà îäíîé ïðÿìîé.
Ïðè ïðîåêòèâíîì ïðåîáðàçîâàíèè àíãàðìîíè÷åñêîå îòíîøåíèå ÷å-
òûðåõ òî÷åê, ëåæàùèõ íà îäíîé ïðÿìîé, íå èçìåíÿåòñÿ.
Ïðîåêòèâíîå ïðåîáðàçîâàíèå îïðåäåëÿåòñÿ ñîîòíîøåíèÿìè:
0 ¯ 0 ¯
1 10 1 10 2 10 3 ¯ 1 10 10 ¯
x = a1 x + a2 x + a3 x
¯ a1 a2 a3 ¯
¯ 0 ¯
0 0
x2 = a21 x1 + a22 x2 + a23 x3
0 0
ãäå ¯ a2 a20 a20 ¯ =
¯ 1 2 3 ¯6 0.
x30 = a30 x1 + a30 x2 + a30 x3 , ¯ 30 30 30 ¯
1 2 3
¯ a1 a2 a3 ¯
Ôîðìóëû ïðîåêòèâíîãî ïðåîáðàçîâàíèÿ ñîáñòâåííûõ òî÷åê ïðîåê-
òèâíîé ïëîñêîñòè â àôôèííûõ êîîðäèíàòàõ ïðèíèìàþò âèä:
0 0 0 0 0 0
a1 x + a12 y + a13 a2 x + a22 y + a23
x = 130
0
0 0 , y = 310
0
0 0 .
a1 x + a32 y + a33 a1 x + a32 y + a33
Ïóñòü çàäàíû ÷åòûðå ïàðû ñîîòâåòñòâåííûõ òî÷åê:
0 0 0
A(a1 : a2 : a3 ) 7→ A0 (a1 : a2 : a3 ) ,
0 0 0
B(b1 : b2 : b3 ) 7→ B 0 (b1 : b2 : b3 ) ,
0 0 0
C(c1 : c2 : c3 ) 7→ C 0 (c1 : c2 : c3 ) ,
52
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