PARAMETRIC SURFACES
Example # 1: Sketch the parametric surface:
x= u y= v z=
2
2
u +v .
Cone
z
x
y
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Example # 2(a.): Find a parametric representation of the
surface: 2 ⋅ z − 3 ⋅ x + 4 ⋅ y = 5 in terms of the parameters: u = x and v = y
.
2⋅ z − 3⋅ u + 4⋅ v = 5
x=u
z ( u , v) =
y=v
5 3
+ ⋅ u − 2⋅ v
2 2
Plane
z
x
y
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Example # 2(b.): Find a parametric representation of the
2
surface: z = x in terms of the parameters: u = x and v = y.
x=u
y=v
z ( u , v) = u
2
Parabolic Cylinder
z
x
y
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Example # 3: Find a parametric representation of the
2
2
cone: z = 3 ⋅ x + 3 ⋅ y in terms of the parameters: ρ and θ, where
( ρ , θ , φ) are spherical coordinates of a point on the surface.
x = ρ ⋅ sin ( φ ) ⋅ cos ( θ )
x + y = ρ ⋅ ( sin ( φ ) )
2
2
2
z = ρ ⋅ cos ( φ ) =
x=
y = ρ ⋅ sin ( φ ) ⋅ sin ( θ )
2
2
3 ⋅ ρ ⋅ sin ( φ )
1
⋅ ρ ⋅ cos ( θ )
2
2
3⋅ x + y =
y=
z = ρ ⋅ cos ( φ )
3 ⋅ ρ ⋅ sin ( φ )
φ = tan
1
⋅ ρ ⋅ sin ( θ )
2
⎞ π
⎜ ⎟= 6
⎝ 3⎠
− 1⎛ 1
z=
3
⋅ρ
2
Example # 4: Eliminate the parameters to obtain an equation in
rectangular coordinates and describe the surface:
→
→
→
→
r( u , v) = i ⋅ ( 3 ⋅ u ⋅ cos ( v) ) + j ⋅ ( 4 ⋅ u ⋅ sin ( v) ) + k ⋅ ( u) for 0 ≤ u ≤ 1 and
0 ≤ v ≤ 2 ⋅ π.
x
= u ⋅ cos ( v)
3
2
y
= u ⋅ sin ( v)
4
2
x
y
2
2
+
=u =z
9
16
2
2
x
y
z =
+
9
16
2
2
z=
0≤ z≤ 1
2
x
y
+
9
16
0≤ z≤ 1
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Elliptical Cone
z
x
y
Example # 5: Find an equation of the tangent plane to the
2
2
parametric surface: x = u, y = v, z = u + v at the point: ( 1 , 2 , 5).
→
→
→
→ 2
2
r( u , v) = i ⋅ ( u) + j ⋅ ( v) + k ⋅ u + v
(
)
→
→
→
∂→
r( u , v) = i ⋅ ( 1) + j ⋅ ( 0) + k ⋅ ( 2 ⋅ u)
∂u
→
→
→
∂→
r( 1 , 2) = i ⋅ ( 1) + j ⋅ ( 0) + k ⋅ ( 2)
∂u
→
→
→
∂→
r( u , v) = i ⋅ ( 0) + j ⋅ ( 1) + k ⋅ ( 2 ⋅ v)
∂v
→
→
→
∂→
r( 1 , 2 ) = i ⋅ ( 0 ) + j ⋅ ( 1 ) + k ⋅ ( 4 )
∂v
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→ ⎡∂→
→
→
⎤ ⎡ ∂→
⎤ →
n = ⎢ r( 1 , 2) ⎥ × ⎢ r( 1 , 2) ⎥ = i ⋅ ( −2) + j ⋅ ( −4) + k ⋅ ( 1)
⎣ ∂u
⎦ ⎣ ∂v
⎦
−2 ⋅ ( x − 1) − 4 ⋅ ( y − 2) + ( z − 5) = 0
z = −5 + 2 ⋅ x + 4 ⋅ y
Surface & Tangent Plane @ (1,2,5)
z
x
y
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