Superficies, Geometria Analitica
-
Upload
choqoferzh -
Category
Engineering
-
view
403 -
download
1
Transcript of Superficies, Geometria Analitica
![Page 1: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/1.jpg)
D
G
V
SUPERFICIES CÓNICAS
![Page 2: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/2.jpg)
Z
Y
X
o
D
G
V
SUPERFICIES CÓNICAS
![Page 3: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/3.jpg)
Z
Y
X
o
D
G
P
vp
C
c
vc
λ vc V
λ)vc(vp:S
SUPERFICIES CÓNICAS
![Page 4: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/4.jpg)
D: parábola en
el plano XZ
p = ¼
V ( 1, 8, -2 )
Z
Y
X
-4
D
V
o
G
![Page 5: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/5.jpg)
D PL YZ
V ( 12, -5, -2 )
Z
Y
X
2
-4
D
V
o
C
G
![Page 6: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/6.jpg)
-5Y
X
2
4
Z
4
2
V
V (-1, 3, 1)
![Page 7: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/7.jpg)
u
SUPERFICIES CILÍNDRICAS
c
Z
Y
X
o
![Page 8: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/8.jpg)
u
SUPERFICIES CILÍNDRICAS
c
Z
Y
X
o
P
C
p
c
ut
utcp:S
![Page 9: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/9.jpg)
4
r=5
Z
X
Y
D
G
(2, 3,4)u
![Page 10: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/10.jpg)
D: hipérbola en el
Plano YZ
b = 2
X
Y
Z
-4 4o
D
u
1,3,4u
![Page 11: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/11.jpg)
-6
105Y
X
2
4
Z
( 4,2,3)u
![Page 12: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/12.jpg)
f(x) = sen x + 2
![Page 13: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/13.jpg)
o
Z
Y
X
SUPERFICIES DE REVOLUCIÓN
![Page 14: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/14.jpg)
o
Z
Y
X
SUPERFICIES DE REVOLUCIÓN
![Page 15: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/15.jpg)
o
Z
Y
X
SUPERFICIES DE REVOLUCIÓN
![Page 16: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/16.jpg)
Y
Z
X
Z
Y
X
![Page 17: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/17.jpg)
o
Z
Y
X
P
p
SUPERFICIES DE REVOLUCIÓN
r
z
x = r cos
C: y =
z = r sen
D: parábola en
el plano YZ
p = 4
416 yzr
![Page 18: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/18.jpg)
4-4 o
Z
Y
X
D
D: hipérbola en el
plano YZ
b = 2
![Page 19: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/19.jpg)
X
Y
Z
-4 4o
D
D: hipérbola en el
plano YZ
b = 2
![Page 20: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/20.jpg)
o
Z
Y
X
![Page 21: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/21.jpg)
Z
Y
X
o
D
G
V
MÉTODO DE LAS GENERATRICES
SUPERFICIES CÓNICAS
![Page 22: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/22.jpg)
X
Y
Z
-4 4o
D
V(-10, -1, 8)
Hipérbola en el plano YZ
b = 3
![Page 23: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/23.jpg)
D PL YZ
V ( 12, -5, -2 )
Z
Y
X
2
-4
D
V
o
C
G
![Page 24: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/24.jpg)
-5Y
X
2
4
Z
4
2
V
V (-1, 3, 1)
![Page 25: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/25.jpg)
u
c
Z
Y
X
o
MÉTODO DE LAS GENERATRICES
SUPERFICIES CILÍNDRICAS
![Page 26: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/26.jpg)
4
r=5
Z
X
Y
D
G
(2, 3,4)u
![Page 27: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/27.jpg)
-6
105Y
X
2
4
Z
( 4,2,3)u
![Page 28: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/28.jpg)
o
Z
Y
X
P
MÉTODO DE LAS GENERATRICES
SUPERFICIES DE REVOLUCIÓN
x 2 + z 2 = 2
G:
y = D: parábola en
el plano YZ
p = 4
![Page 29: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/29.jpg)
ELIPSE // AL PLANO XY
Gira alrededor de su eje focal
4
10
Z
X
Y
D
G
6
![Page 30: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/30.jpg)
X
Y
Z
-4
4
o
D
D
Directriz:
hipérbola en
el plano YZ,
b = 2
![Page 31: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/31.jpg)
o
Z
Y
X
![Page 32: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/32.jpg)
o
Z
Y
X C(4, 12, 8)
R = 6
C
![Page 33: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/33.jpg)
D
GD
ELIPSOIDE ELÍPTICO
![Page 34: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/34.jpg)
D1 ELIPSE // AL PLANO XY
D2 ELIPSE // AL PLANO YZ
10
Z
X
Y
6
4D1
G
D2
![Page 35: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/35.jpg)
PARABOLOIDE ELÍPTICO
![Page 36: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/36.jpg)
SUPERFICIES CUÁDRICAS O CUADRÁTICAS
A x 2 + B x y + C y 2 + D x z + E z 2 + F y z + G x + H y + I z + J = 0
![Page 37: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/37.jpg)
12
2
2
2
2
2
c
lz
b
ky
a
hx1º - Los tres signos positivos indican que se trata de un
Elipsoide con centro C ( h, k, l ), si a = b ó a = c ó
b = c es de revolución.
2º - Un signo negativo indica que se trata de un
Hiperboloide Elíptico de un manto con centro
C (h, k, l ), si a = b ó a = c ó b = c puede ser de
revolución.
3º - Dos signos negativos indican que se trata de un
Hiperboloide Elíptico de dos mantos con centro
C ( h, k, l ), si a = b ó a = c ó b = c puede ser de
revolución.
22
2
2
2
c
lz
b
ky
a
hx Paraboloide Elíptico con vértice en V ( h, k, l ), si
a = b es de revolución.
2
2
2
2
2
2
c
lz
b
ky
a
hx Cono Elíptico con vértice en V ( h, k, l )
si a = b es de revolución.
IDENTIFICACIÓN DE SUPERFICIES POR SU ECUACIÓN CARTESIANA
![Page 38: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/38.jpg)
Z
Y
X
Curvas de nivel
![Page 39: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/39.jpg)
Z
Y
X
Curvas de nivel
![Page 40: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/40.jpg)
Z
Y
X
Curvas de nivel
![Page 41: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/41.jpg)
Curvas de nivel
Superficie
Mapa de contorno
![Page 42: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/42.jpg)
![Page 43: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/43.jpg)
![Page 44: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/44.jpg)
![Page 45: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/45.jpg)
![Page 46: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/46.jpg)
4
5
Z
X
Y
6
ELIPSOIDE DE
REVOLUCIÓN
S: 36 x2 + 36 y2 + 25 z2 - 288 x - 360 y - 300 z + 1476 = 0
2 2 2
4 5 61
25 25 36
x y z
![Page 47: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/47.jpg)
4-4
Z
Y
X
-4
4
S: x2 + y2 – 4 z2 -16 = 0
HIPERBOLOIDE DE REVOLUCIÓN
DE UN MANTO
2 2 2
116 16 4
x y z
![Page 48: Superficies, Geometria Analitica](https://reader034.fdocuments.net/reader034/viewer/2022042607/55b13436bb61eb56248b46fd/html5/thumbnails/48.jpg)
o
Z
Y
X
2
5
-3
S: 2 x2 + 2 y2 + 12 x - 20 y – 8 z + 84 = 0
PARABOLOIDE DE REVOLUCIÓN
6
2 2
3 5 4 2x y z