Аналитическая геометрия. Часть III. Многомерные пространства. Гиперповерхности второго порядка. Шурыгин В.В. - 62 стр.

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w
1
= z
1
w
2
= z
2
−
1
2
w
3
= z
3
−
3
2
(w
1
)
2
+ (w
2
)
2
−(w
3
)
2
−3 = 0
x
1
0
=
√
3w
1
x
2
0
=
√
3w
2
x
3
0
=
√
3w
3
3
(x
1
0
)
2
+ (x
2
0
)
2
− (x
3
0
)
2
= 1.
x
i
0
w
i
z
i
y
i
x
i
x
1
0
=
√
3(x
1
− x
2
+ 1),
x
2
0
=
√
3
2
(x
2
+ x
3
− 1),
x
3
0
=
√
3
2
(x
2
− x
3
− 3).
A
3
`
1
`
2
`
3
Φ `
`
1
`
2
`
3
`
1
r = r
1
+ t
1
a
1
`
2
r = r
2
+ t
2
a
2
`
1
⊂ π
1
r = r
1
+ λ
1
a
1
+ λ
2
a
2
`
2
⊂ π
2
r = r
2
+ λ
1
a
1
+ λ
2
a
2
`
1
`
2
`
3
V
3
A
3
A
3
`
1
x
2
= x
3
= 0 `
2
x
1
= 0 x
3
= 1
`
3
x
1
= 1 x
2
= 1
’-/6* .3*-+35E-B5,08 1--3D0,57 w1 = z1  w2 = z2 − 21  w3 = z3 − 23 235B
,*,0* .30,0:5*7 B0D (w1)2 + (w2)2 − (w3)2 − 3 = 0 < ’-65958 -1-,)57*6>,-
  0  √        0   √         0  √                               )
x1 = 3w1  x2 = 3w2  x3 = 3w3  .-/6* D*6*,08 ,5 3 .-62 5*: 15,-,0
)*/1-* 235,*,0*
                            0             0
                        (x1 )2 + (x2 )2 − (x3 )2 = 1.
                                                       0               •·I‹
  35B,*,0* •·I‹ 8B68*7/8 235B,*,0*: -D,-.-6-/7,-9- 90.*3+-6-0D5< A35
“58 .-/6*D-B57*6>,- xi )*3*E wi  E57*: )*3*E zi  yi 0 xi  .-62)0: .3*-+
                        0


35E-B5,0* 1--3D0,57 .30B-D8ˆ** 235B,*,0* •··‹ 1 B0D2 •·I‹~
                                 0   √
                            x1 =         3(x1 − x2 + 1),
                                     √
                                20     3 2
                            x =      2 (x         + x3 − 1),
                                     √
                                 0     3 2
                            x3 =     2 (x         − x3 − 3).
   ªœZœ«œ –Û^ 5„„0,,-: .3-/735,/7B* A D5,A 730 .-.53,- /13*ˆ0B5
‡ˆ0*/8 .38:A* `1  `2 0 `3  ,5.35B68‡ˆ0* B*17-3A 1-7-3AC 60,*”,- ,*E5
                                           3


B0/0:A < -15E57>  )7- :,-“*/7B- Φ 7-)*1  .30,5D6*“5ˆ0C .38:A: `  .*
3*/*15‡ˆ0: -D,-B3*:*,,- B/* 730 .38:A* `1  `2 0 `3  .3*D/75B68*7 /-+-”
-D,-.-6-/7,A” 90.*3+-6-0D <
   ÈYÉY\]Y^ 13*ˆ0B5‡ˆ0*/8 .38:A* ` / 235B,*,0*: r = r + t1a 0 ` /
235B,*,0*: r = r2 + t2a2 6*“57 B .53566*6>,AC .6-/1-/78C `1 ⊂ π1 / 235B
                                       1                     1       1   2


,*,0*: r = r1 + λ1a1 + λ2a2 0 `2 ⊂ π2 / 235B,*,0*: r = r2 + λ1a1 + λ2a2 <
  30 .53A 7510C .6-/1-/7*” D68 D5,,AC .38:AC `1  `2 0 `3 -935,0)0B5‡7
.53566*6*.0.*D < ’30,0:58 -D,2 0E B*3‰0, ¨7-9- .53566*6*.0.*D5 E5 ,5
)56- 1--3D0,57 5 B*17-3A *9- 3*+*3 E5 +5E0/ B V  5//-½003-B5,,-: / A 
.-62)0: 3*.*3 B A3 < /0/7*:* 1--3D0,57 -.3*D*68*:-” ¨70: 3*.*3-:
                                                3                          3


D5,,A* .38:A* +2D27 0:*7> 235B,*,08 `1 ~ x2 = x3 = 0  `2 ~ x1 = 0  x3 = 1 
5 `3 ~ x1 = 1  x2 = 1 <




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