povrŠi u prostoru -...
TRANSCRIPT
![Page 1: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/1.jpg)
POVRŠI U PROSTORU
Površ u prostoru može biti zadata jednačinama : U parametarskom obliku U eksplicitnom obliku
),(1 vux ),(2 vuy
),(3 vuz
),( yxfz
21
21
yyyxxx
21
21
vvvuuu
U implicitnom obliku
0),,( zyxF
![Page 2: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/2.jpg)
x
y
POVRŠI U PROSTORU )sin( yxz
10101010
yx
Eksplicitni oblik ‐ primer
![Page 3: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/3.jpg)
POVRŠI U PROSTORU
yxz sinsin
10101010
yx
Eksplicitni oblik ‐ primeri
22 yxz
2222
yx
2222
yx
)( 22 yxez
![Page 4: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/4.jpg)
POVRŠI U PROSTORU 0coscos zycox
10101010
yx
Implicitni oblik ‐ primer
Schwartz‐ova površ
![Page 5: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/5.jpg)
POVRŠI U PROSTORU 0sincossincossin xzzyycox
4444
yx
Implicitni oblik ‐ primer
Giroid
![Page 6: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/6.jpg)
POVRŠI U PROSTORU
Površi drugog reda
0222 QPzNyMxFxzEyzDxyCzByAx
Svodjenjem na kanonski oblik dobijaju se: Elipsoid I kao poseban slučaj sfera Jednograni hiperboloid Dvograni hiperboloid Eliptički paraboloid Hiperbolički paraboloid
![Page 7: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/7.jpg)
POVRŠI U PROSTORU
Površi drugog reda
Preseci sa ravnima paralelnim sa yz‐ravni: su krive drugog reda
i
iiii
pxQPzNyMpzFpEyzyDpCzByAp 0222
ipx
0222 QPzNyMxFxzEyzDxyCzByAx
![Page 8: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/8.jpg)
POVRŠI U PROSTORU
Površi drugog reda
Preseci sa ravnima paralelnim sa xz‐ravni: su krive drugog reda
iqy
i
iiii
qyQPzNqMxFxzzEqDxqCzBqAx 0222
0222 QPzNyMxFxzEyzDxyCzByAx
![Page 9: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/9.jpg)
POVRŠI U PROSTORU
Površi drugog reda
0222 QPzNyMxFxzEyzDxyCzByAx
Preseci sa ravnima paralelnim sa xy‐ravni: su krive drugog reda
irz
i
iiii
rzQNyMxFxrEyrDxyCrByAx 0Pr222
![Page 10: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/10.jpg)
POVRŠI U PROSTORU
Površi drugog reda ‐ elipsoid
0222 QPzNyMxFxzEyzDxyCzByAx
Elipsoid: 12
2
2
2
2
2
cz
by
ax
Sfera: 2222 rzyx
220
20
20 )()()( rzzyyxx
a, b, c ‐ poluose elipsoida czcbybaxa
rcba Poseban slučaj
![Page 11: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/11.jpg)
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
Površi drugog reda ‐ elipsoid
ipx
12
2
2
2
2
2
cz
by
api
2
2
2
2
2
2
1ap
cz
by i
czcbybaxa
111 2
22
2
2
22
2
apc
z
apb
y
ii
ipx
![Page 12: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/12.jpg)
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
Površi drugog reda ‐ elipsoid
iqy czcbybaxa
1
2
2
2
2
2
2
cz
bq
ax i
2
2
2
2
2
2
1bq
cz
ax i
111 2
22
2
2
22
2
bqc
z
bqa
x
ii
iqy
![Page 13: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/13.jpg)
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
Površi drugog reda ‐ elipsoid
irz czcbybaxa
1
2
2
2
2
2
2
cr
by
ax i
2
2
2
2
2
2
1cr
by
ax i
111 2
22
2
2
22
2
crb
y
cra
x
ii
irz
![Page 14: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/14.jpg)
POVRŠI U PROSTORU
Površi drugog reda ‐ elipsoid
![Page 15: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/15.jpg)
Površi drugog reda ‐ jednograni hiperboloid
Jednograni hiperboloid: 12
2
2
2
2
2
cz
by
ax
a, b, c ‐ poluose hiperboloida
Oz ‐ imaginarna osa jednogranog hiperboloida i on sa njom nema zajedničkih tačaka
12
2
2
2
2
2
cz
by
ax
12
2
2
2
2
2
cz
by
ax
Oy ‐ imaginarna osa
Ox ‐ imaginarna osa
POVRŠI U PROSTORU
c – imaginarna poluosa
![Page 16: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/16.jpg)
Površi drugog reda ‐ jednograni hiperboloid
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
ipx
12
2
2
2
2
2
cz
by
api
2
2
2
2
2
2
1ap
cz
by i
111 2
22
2
2
22
2
apc
z
apb
y
ii
ipx
![Page 17: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/17.jpg)
Površi drugog reda ‐ jednograni hiperboloid
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
iqy
12
2
2
2
2
2
cz
bq
ax i
2
2
2
2
2
2
1bq
cz
ax i
111 2
22
2
2
22
2
bqc
z
bqa
x
ii
iqy
![Page 18: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/18.jpg)
Površi drugog reda ‐ jednograni hiperboloid
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
irz
12
2
2
2
2
2
cr
by
ax i
2
2
2
2
2
2
1cr
by
ax i
111 2
22
2
2
22
2
crb
y
cra
x
ii
irz
![Page 19: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/19.jpg)
Površi drugog reda ‐ jednograni hiperboloid
POVRŠI U PROSTORU
![Page 20: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/20.jpg)
Površi drugog reda ‐ dvograni hiperboloid
Dvograni hiperboloid: 12
2
2
2
2
2
cz
by
ax
a, b, c ‐ poluose hiperboloida
POVRŠI U PROSTORU
a, b ‐ imaginarne poluose
12
2
2
2
2
2
cz
by
ax 1
2
2
2
2
2
2
cz
by
ax
![Page 21: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/21.jpg)
Površi drugog reda ‐ dvograni hiperboloid
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
12
2
2
2
2
2
cz
by
api
2
2
2
2
2
2
1ap
cz
by iipx
2
2
2
2
2
2
1ap
by
cz i
111 2
22
2
2
22
2
apb
y
apc
z
ii
ipx
![Page 22: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/22.jpg)
Površi drugog reda ‐ dvograni hiperboloid
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
iqy
12
2
2
2
2
2
cz
bq
ax i
2
2
2
2
2
2
1bq
cz
ax i
2
2
2
2
2
2
1bq
ax
cz i
2
2
2
22
2
2
22
2
111
bq
bqa
x
bqc
z i
ii
iqy
![Page 23: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/23.jpg)
Površi drugog reda ‐ dvograni hiperboloid
12
2
2
2
2
2
cz
by
ax
POVRŠI U PROSTORU
irz
12
2
2
2
2
2
cr
by
ax i
2
2
2
2
2
2
1cr
by
ax i
111 2
22
2
2
22
2
crb
y
cra
x
ii
irz
![Page 24: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/24.jpg)
Površi drugog reda ‐ eliptički paraboloid
Eliptički paraboloid: zqy
px 2
22
POVRŠI U PROSTORU
yqz
px 2
22
xqz
py 2
22
![Page 25: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/25.jpg)
Površi drugog reda ‐ eliptički paraboloid
zqy
px 2
22
POVRŠI U PROSTORU
ipx z
qy
ppi 2
22
ppz
qy i
22
2
ppzqy i
22
22
ipx
![Page 26: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/26.jpg)
Površi drugog reda ‐ eliptički paraboloid
zqy
px 2
22
POVRŠI U PROSTORU
iqy zq
qpx i 2
22
qqz
px i
22
2
qqzpx i
22
22
iqy
![Page 27: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/27.jpg)
Površi drugog reda ‐ eliptički paraboloid
zqy
px 2
22
POVRŠI U PROSTORU
irz irq
ypx 2
22
122
22
ii qr
yprx
irz
![Page 28: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/28.jpg)
Površi drugog reda ‐ eliptički paraboloid
POVRŠI U PROSTORU
![Page 29: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/29.jpg)
Površi drugog reda ‐ hiperbolički paraboloid
POVRŠI U PROSTORU
Hiperbolički paraboloid: zqy
px 2
22
xqz
py 2
22
yqz
px 2
22
![Page 30: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/30.jpg)
Površi drugog reda ‐ hiperbolički paraboloid
POVRŠI U PROSTORU
zqy
px 2
22
ipx z
qy
ppi 2
22
ppz
qy i
22
2
ppzqy i
22
22
ipx
![Page 31: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/31.jpg)
Površi drugog reda ‐ hiperbolički paraboloid
POVRŠI U PROSTORU
zqy
px 2
22
iqy zq
qpx i 2
22
qqz
px i
22
2
qqzpx i
22
22
iqy
![Page 32: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/32.jpg)
Površi drugog reda ‐ hiperbolički paraboloid
POVRŠI U PROSTORU
zqy
px 2
22
irz
irqy
px 2
22
iii qr
yprx 1
22
22
irz
![Page 33: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/33.jpg)
Površi drugog reda ‐ hiperbolički paraboloid
POVRŠI U PROSTORU
![Page 34: POVRŠI U PROSTORU - elearning.rcub.bg.ac.rselearning.rcub.bg.ac.rs/moodle/pluginfile.php/27459/mod_resource... · Površi drugog reda ... su krive drugog reda i i i i i x p Ap2 By2](https://reader030.vdocuments.pub/reader030/viewer/2022040708/5e0a0f0192cfed256f3ab9cc/html5/thumbnails/34.jpg)
Površi drugog reda ‐ hiperbolički paraboloid
POVRŠI U PROSTORU