Cuadrice - math.uaic.rooanacon/depozit/Curs-cuadrice.pdf · Oana Constantinescu Cuadrice. Cilindrul...
Transcript of Cuadrice - math.uaic.rooanacon/depozit/Curs-cuadrice.pdf · Oana Constantinescu Cuadrice. Cilindrul...
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Cuadrice
Oana Constantinescu
March 19, 2013
Oana Constantinescu Cuadrice
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De�nition
O cuadrica este locul geometric al punctelor din spatiu ale caror
coordonate in raport cu un reper ortonormat R = {O; i , j , k}veri�ca o ecuatie de tipul:
a11x2 + a22y
2 + a33z2 + 2a12xy + 2a13xz + 2a23yz + (1)
2a10x + 2a20y + 2a30z + a00 = 0 ,
a211 + a222 + a233 > 0.
tXAX + 2BX + a00 = 0, A =
a11 a12 a13a21 a22 a23a31 a32 a33
6= O3 (2)
A =t A, B = (a10 a20 a30) .
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Theorem
Calitatea unei submultimi de a � o cuadrica nu depinde de reperul
ortonormat in raport cu care s-a dat ecuatia ei.
La o schimbare de repere ortonormate
R = {O; i , j , k} → R′ = {O ′; i ′, j ′, k ′}, ce determina
schimbarea de coordonate X = SX ′ + S0, ecuatia (2) devine
tX ′A′X ′ + 2B ′X ′ + a′00 = 0,
unde
A′ =t SAS 6= O3, B ′ = (tS0A + B) S a′00 = f (S0) .
![Page 4: Cuadrice - math.uaic.rooanacon/depozit/Curs-cuadrice.pdf · Oana Constantinescu Cuadrice. Cilindrul eliptic x 2 a 2 + y 2 b 2 1 = 0 Oana Constantinescu Cuadrice. Cilindrul hiperbolic](https://reader031.fdocuments.net/reader031/viewer/2022020306/5e20ff021125d375182755e5/html5/thumbnails/4.jpg)
Elipsoidul
x2
a2+
y2
b2+
z2
c2− 1 = 0 (3)
Oana Constantinescu Cuadrice
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Hiperboloidul cu o panza
x2
a2+
y2
b2− z2
c2− 1 = 0 (4)
Oana Constantinescu Cuadrice
![Page 6: Cuadrice - math.uaic.rooanacon/depozit/Curs-cuadrice.pdf · Oana Constantinescu Cuadrice. Cilindrul eliptic x 2 a 2 + y 2 b 2 1 = 0 Oana Constantinescu Cuadrice. Cilindrul hiperbolic](https://reader031.fdocuments.net/reader031/viewer/2022020306/5e20ff021125d375182755e5/html5/thumbnails/6.jpg)
Generatoarele hiperboloidului cu o panza
Hiperboloidul cu o panza este o cuadrica riglata, adica exista
o familie de drepte cu proprietatile:
1 orice dreapta din familie este situata pe cuadrica;2 prin orice punct al cuadricei trece cel putin o dreapta din
familie.
O astfel de familie se numeste sistem de generatoare
rectilinii pentru cuadrica respectiva.
(xa− z
c
) (xa
+ zc
)=(1− y
b
) (1 + y
b
)dλ :
{xa− z
c= λ
(1− y
b
)λ(xa
+ zc
)=(1 + y
b
)δµ :
{xa− z
c= µ
(1 + y
b
)µ(xa
+ zc
)=(1− y
b
) λ, µ ∈ R ∪ {∞}
Oana Constantinescu Cuadrice
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Hiperboloidul cu doua panze
x2
a2− y2
b2− z2
c2− 1 = 0 (5)
Oana Constantinescu Cuadrice
![Page 8: Cuadrice - math.uaic.rooanacon/depozit/Curs-cuadrice.pdf · Oana Constantinescu Cuadrice. Cilindrul eliptic x 2 a 2 + y 2 b 2 1 = 0 Oana Constantinescu Cuadrice. Cilindrul hiperbolic](https://reader031.fdocuments.net/reader031/viewer/2022020306/5e20ff021125d375182755e5/html5/thumbnails/8.jpg)
Paraboloidul eliptic
x2
a2+
y2
b2− 2z = 0 (6)
Oana Constantinescu Cuadrice
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Paraboloidul hiperbolic
x2
a2− y2
b2− 2z = 0 (7)
-2 -1 0 1 2
-1-0.500.51
-1
0
1
2
3
-1-0.5
Oana Constantinescu Cuadrice
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Generatoarele paraboloidului hiperbolic
(xa
+ yb
) (xa− y
b
)= 2z
dλ :
{xa
+ yb
= 2λ
λ(xa− y
b
)= z
δµ
{xa
+ yb
= µz
µ(xa− y
b
)= 2
λ, µ ∈ R ∪ {∞}, µ 6= 0.
Oana Constantinescu Cuadrice
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Cilindri patratici
Se considera o conica (γ), reprezentata intr-un reper
ortonormat din spatiu prin{a11x
2 + 2a12xy + a22y2 + 2a10x + 2a20y + a00 = 0
z = 0
De�nition
Locul geometric al punctelor dreptelor din spatiu δ, paralele cu axa
Oz a reperului considerat, care se sprijina pe conica (γ), senumeste cilindru patratic. Conica (γ) se numeste curba
directoare iar dreptele paralele cu Oz se numesc generatoarele
(rectilinii) ale cilindrului.
Ecuatia unui astfel de cilindru patratic este
a11x2 + 2a12xy + a22y
2 + 2a10x + 2a20y + a00 = 0 (8)
Oana Constantinescu Cuadrice
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Cilindrul eliptic
x2
a2+
y2
b2− 1 = 0
Oana Constantinescu Cuadrice
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Cilindrul hiperbolic
x2
a2− y2
b2− 1 = 0
Oana Constantinescu Cuadrice
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Cilindrul parabolic
y2 = 2px
Oana Constantinescu Cuadrice
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Cilindru general (nu e cuadrica)
Oana Constantinescu Cuadrice
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Conuri patratice
Se considera o conica (γ), reprezentata intr-un reper
ortonormat din spatiu prin{a11x
2 + 2a12xy + a22y2 + 2a10x + 2a20y + a00 = 0
z = k 6= 0
De�nition
Locul geometric al punctelor dreptelor din spatiu δ, care se sprijina
pe conica (γ) si trec toate prin O, se numeste con patratic (cu
varful in O). Conica (γ) se numeste curba directoare iar dreptele
paralele cu OM, M ∈ γ se numesc generatoarele (rectilinii) ale
conului.
Ecuatia acestui con patratic este
a11k2x2 + a22k
2y2 + a00z2 + 2a12k
2xy + 2a10kxz + 2a20kyz = 0
Oana Constantinescu Cuadrice
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Con patratic
x2
a2+
y2
b2− z2
c2= 0
Oana Constantinescu Cuadrice
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Con general (nu e cuadrica)
Oana Constantinescu Cuadrice
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Conicele ca sectiuni in conul de rotatie
Oana Constantinescu Cuadrice
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Cuadrice degenerate
O pereche de plane
(ax + by + cz + d)(a′x + b′y + c ′z + d ′) = 0, a2 + b2 + c2 > 0
a′2 + b′2 + c ′2 > 0
O dreapta dubla
x2 + y2 = 0
Un punct dublu
x2 + y2 + z2 = 0
O cuadrica vida
x2 + y2 + z2 + 1 = 0
Oana Constantinescu Cuadrice
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Invarianti ortogonali si centro-ortogonali ai unei cuadrice
Fie o cuadrica (Γ) de�nita, in raport cu un reper ortonormat �xat,
prin ecuatia
tXAX + 2BX + a00 = 0, A =
a11 a12 a13a21 a22 a23a31 a32 a33
6= O3
A =t A, B = (a10 a20 a30) ,
D =
(A tB
B a00
)=t D.
De�nim:
I = Tr(A), δ = det(A), J =suma minorilor diagonali de
ordinul 2 lui A
∆ = det(D), L = suma minorilor diagonali de ordinul 2 ai lui
D, K = suma minorilor diagonali de ordinul 3 ai lui D
Oana Constantinescu Cuadrice
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Theorem
a) δ, ∆, I , J sunt invarianti ortogonali ai cuadricei.
b) L, K sunt invarianti centro-ortogonali ai cuadricei.
c) Daca δ = ∆ = 0, atunci K este invariant ortogonal.
d) Daca δ = ∆ = K = J = 0, atunci Leste invariant ortogonal.
Fie λ1, λ2, λ3 valorile proprii (nu neaparat distincte) ale matricii A:
λ3 − Iλ2 + Jλ− δ = 0.
Putem obtine o clasi�care a cuadricelor in functie de semnele
invariantilor asociati acesteia cat si a valorilor proprii.
Oana Constantinescu Cuadrice
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Clasi�carea cuadricelor
δ ∆ λ1, λ2, λ3 − ∆λi δ
K L Cuadrica
> 0 6= 0 + + + Elipsoid
< 0 6= 0 + + - Hiperboloid cu o panza
> 0 6= 0 - - + Hiperboloid cu doua panze
< 0 6= 0 - - - Cuadrica vida
6= 0 0 acelasi semn Punct dublu
6= 0 0 + + - Con patratic
0 6= 0 λ1λ2 > 0, λ3 = 0 Paraboloid eliptic
0 6= 0 λ1λ2 < 0, λ3 = 0 Paraboloid hiperbolic
0 0 + + 0 > 0 Cuadrica vida
0 0 λ1λ2 > 0, λ3 = 0 0 Dreapta dubla
0 0 + + 0 < 0 Cilindru eliptic
0 0 - - 0 > 0 Cilindru eliptic
0 0 - - 0 < 0 Cuadrica vida
0 0 λ1λ2 < 0, λ3 = 0 6= 0 Cilindru hiperbolic
Oana Constantinescu Cuadrice
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δ ∆ λ1, λ2, λ3 − ∆λi δ
K L Cuadrica
0 0 λ1λ2 < 0, λ3 = 0 0 Plane secante
0 0 λ1 6= 0, λ2 = λ3 = 0 0 > 0 Cuadrica vida
0 0 λ1 6= 0, λ2 = λ3 = 0 0 0 Plan dublu
0 0 λ1 6= 0, λ2 = λ3 = 0 0 < 0 Plane paralele
0 0 λ1 6= 0, λ2 = λ3 = 0 6= 0 Cilindru parabolic
Oana Constantinescu Cuadrice
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Gaudi
Oana Constantinescu Cuadrice
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Oana Constantinescu Cuadrice
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Oana Constantinescu Cuadrice
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Oana Constantinescu Cuadrice
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Oana Constantinescu Cuadrice
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Oana Constantinescu Cuadrice