Section 10.5
Parametric Surfaces
(1) Surfaces,
(2) Grid Lines,
(3) Parameterizing Surfaces.
MATH 122 (Section 10.5)
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Surfaces
Consider a vector function ~r (u, v ) of two variables
~r (u, v ) = hf (u, v ), g (u, v ), h(u, v )i
with points (u, v ) in the domain D which lie in a uv -plane. The set of
points satisfying the equation in R3
S = {(f (u, v ), g (u, v ), h(u, v )) | (u, v ) ∈ D}
is called a parametric surface.
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If a parametric surface S is given by a vector function ~r (u, v ), the surface
can be sketched using the grid lines of the domain. For a constant c
there are two space curves,
(I) u~c (t) = ~r (c, t)
(II) ~vc (t) = ~r (t, c)
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Example: Sketch and identify the surface parameterized by the vector
function
~r (u, v ) = hu cos(v ), u sin(v ), ui
Solution: Take the parametric equation of the surface
x = u cos(v )
y = u sin(v )
z =u
Eliminate u and v to obtain an
equation in (x, y , z).
x = z cos(v )
y = z sin(v )
x 2 +y 2 = (z cos(v ))2 +(z sin(v ))2 = z 2
The parametric surface is a cone.
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Example: Find a vector equation for the plane which passes through the
point P with position vector r~0 and contains two non-parallel vectors ~a and
~b.
Solution: Any point Q on the plane
can be found through combinations
of multiples of the vectors ~a and ~b.
~r (u, v ) = r~0 + u~a + v ~b
Example: Find a parametrization for the ellipsoid
x2 y2 z2
+ 2 + 2 =1
a2
b
c
Solution: We can use spherical coordinates to parameterize the ellipsoid
~r (θ, φ) = ha sin(φ) cos(θ), b sin(φ) cos(θ), c cos(φ)i
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Example: Find a parametrization for the hyperboloid of one sheet
x2 y2 z2
+ 2 − 2 =1
a2
b
c
Solution: There are infinitely many parameterizations of any surface. We
will find two for this surface:
(I) Set z = u, then
x2
a2
+
y2
b2
=1+
u2
.
c2
We can then let
r
x =a
u2
1 + 2 cos(v )
c
(II) Let z = c tan(u), then
x=
MATH 122 (Section 10.5)
x2
a2
+
y2
b2
=
a
cos(v )
cos(u)
r
y =b
1
.
cos2 (u)
y=
Parametric Surfaces
1+
u2
sin(v )
c2
We can then let
b
sin(v )
cos(u)
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Surface of Revolution
By revolving a planar curve y = f (x) about the z-axis, we create a surface
of revolution with z-traces of the form x 2 + y 2 = f (z)2 . This surface can
be parameterized by the vector function
~r (θ, z) = hf (z) cos(θ), f (z) sin(θ), zi
Example: The surface resulting from
revolving f (z) = z 2 about the z-axis
~r (θ, z) = hz 2 cos(θ), z 2 sin(θ), zi
Rotating the planar curve about the x-axis has a parametrization
x =x
y = f (x) cos(θ)
z = f (x) sin(θ)
Rotating the planar curve about the y -axis has parametrization
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