z = f (x, y ) x, y
x = u y = v z = f (u, v)
φ(u, v) = (u , v , f (u, v))
(1, 0, 1), (−2, 3, 0)
π
(−2, 3, 0) − (1, 0, 1) = (−3, 3, −1)
(4, 1, −2)
(4, 1, −2) − (1, 0, 1) = (3, 1, −3)
i
n = (−3, 3, −1)×(3, 1, −3) =
j j
k
3 3 −1 = ( −8, −12, −12) 3 1 −3 (−3, 3, −1), (3, 1, −3)
π = (P − P 0 ) · n = 0 π = ((x , y , z ) − (1, 0, 1)) · (−8, −12, −12) = 0 ⇒
z =
20 + 12y + 8 x −12
−
∴
−
8x − 12y − 12z + 20 = 0
⇒ −
φ(u, v) =
u,v,
20 + 12v + 8 u −12
−
XZ
Z
XZ
(α(v ), β (v )) v
∈
[a, b]
XZ
Z
P
cos(u) =
x y , sen(u) = , z = β (v ) α(v ) α(v )
x = cos(u)α(v ), y = sen(u)α(v ), z = β (v )
u
f (u, v ) XZ f (u, v) = (cos( u)α(v ), sen(u)α(v ), β (v )) u
∈
[0, 2π ], v ∈ [a, b] C 1
r C 1
C 1
C 2
(R + r cos(v ), r sen(v ))
f (u, v) = ((R + r cos(v )) cos(u), (R + r cos(v )) sen(u), r sen(v ))
C 2
γ (t) = (x(t), 0, z (t))
(v sen
u
, v cos
2
u
u ∈ [0, 2π ],
2
v
[ 1, 1]
∈ −
π
f (u, v) =
1 + v sen
1 + v sen
u
2
u
, v cos
2
u
2
cos(u), 1 + v sen
v
u
2
[ 1, 1],
∈ −
sen(u), v cos
u
∈
[0, 2π ]
u
2
v
[ 1, 1],
∈ −
u ∈ [0, 2π ]
f (A) = f (0, −1) = (1, 0, −1) = f (2π, 1)
f (B ) = (0, 1) = (1, 0, 1) = f (2π, −1)
XZ
X
XZ
(α(v ), β (v )) v
∈
[a, b]
u ∈ [0, 2π ]
P
cos(u) =
z y , sen(u) = β (v ) β (v )
P
(x , y , z ) = (α(v ), β (v ) sen(u), β (v ) cos(u)) f (u, v)
f (u, v ) = (α(v ), β (v ) sen(u), β (v ) cos(u))
