Аналитическая геометрия. Часть III. Многомерные пространства. Гиперповерхности второго порядка. Шурыгин В.В. - 7 стр.

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π
n1
( ) P Q π
n1
[P Q] π
n1
: [P Q]π
n1
=
P Q
π
n1
π
n1
A
n
f : A
n
R f(M) = b
1
x
1
M
+ . . . + b
n
x
n
M
+ b
n+1
P Q π
n1
sgn(f(P )) = sgn(f(Q))
f(M)
π
m
π
0
k
A
n
π
m
π
0
k
6=
π
m
= {M
0
, L
m
} π
0
k
= {M
0
0
, L
0
k
}
A
n
π
m
π
0
k
3 A
π
m
π
0
k
= {A, L
m
L
0
k
}
B π
m
π
0
k
AB L
m
, L
0
k
AB L
m
L
0
k
π
m
= {M
0
, L
m
} π
0
k
= {M
0
0
, L
0
k
}
r = r
0
+ t
α
a
α
α = 1, . . . , m r =
r
0
0
+ u
a
b
a
a = 1, . . . , k M π
m
π
0
k
{t
α
} {u
a
} r
0
+ t
α
a
α
= r
0
0
+ u
a
b
a
M
t
α
a
α
u
a
b
a
= r
0
0
r
0
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   VWXYZY[Y\]Y^ ž}hi‚ njbj xaemfeoghvghi‚ π € Ä}nmd xgkgfai‚w sig
nkm (fjloasbcm) igsva P a Q omyji eg gnb} higfgb}
                                               n−1
                                                        gi πn−1 w mhoa gip
fmlgv [P Q] bm hgnmfyai igsmvw efabjnomyjqar πn−1 : [P Q]∩πn−1 = ∅ € ¹
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gi πn−1 €
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D*60: „2,1½0‡ f : An → R „-3:26-” f (M ) = b1x1M + . . . + bnxnM + bn+1 <
                                       n−1   n


  -15E57>  )7- 7-)10 P 0 Q 6*“57 .- -D,2 /7-3-,2 -7 πn−1 7-9D5 0 7-6>1-
7-9D5 1-9D5 sgn(f (P )) = sgn(f (Q)) < 510: -+35E-: 2 90.*3.6-/1-/70 B
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’2/7> E5D5,A DB* .6-/1-/70 πm 0 πk0 B 5„„0,,-/ .3-/735,/7B* An < 5/
/:-730: B-E:-“,A* /62)50 0C BE50:,-9- 35/.-6-“*,08 <
    –^ π ∩ π0 6= ∅ <
    ŽXYZ[—¡Y\]Y^        Æhoa eoghvghia πm = {M0, Lm} a πk0 = {M00 , L0k } k efgp
        m       k


hifjbhikm An adm{i bme}higm emfmhmsmbamw ig gba emfmhmvj{iht eg
eoghvghia€ _ admbbgw mhoa πm ∩ πk0 3 A w ig πm ∩ πk0 = {A, Lm ∩ L0k } €
    £—™œ¤œšY[¥˜š›—^ *”/7B07*6>,-  B ∈ π ∩ π0 ⇐⇒ −             →
                                                              AB ∈ Lm , L0k ⇐⇒
                                                   m     k
AB ∈ Lm ∩ L0k < 
−→

    ’2/7> .6-/1-/70 πm = {M0, Lm} 0 πk0 = {M00 , L0k } E5D5,A  /--7B*7/7B*,
,-  .535:*730)*/10:0 235B,*,08:0 r = r0 + tαaα  α = 1, . . . , m  0 r =
                                                        )
r00 + ua ba  a = 1, . . . , k < 5D02/ B*17-3 B/81-” 7- 10 M ∈ πm ∩ πk0 0:**7 DB5
,5+-35 B,273*,,0C 1--3D0,57 {tα} 0 {ua} < ’30 ¨7-: r0 + tαaα = r00 + uaba <
7/‡D5 /6*D2*7 )7- B,273*,,0* 1--3D0,57A 7-)10 M 2D-B6*7B-38‡7 B*1
7-3,-:2 235B,*,0‡
                                α  tα a − ua b = r0 − r
                                        a     0    0
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