Section 16.6 Parametric Surfaces and Their Areas
Transcript of Section 16.6 Parametric Surfaces and Their Areas
Section 16.6 Parametric Surfaces and Their Areas
Xin Li
MAC2313 Summer 2020
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 1 / 20
parametric curve and surface
Recall: When we parameterized a curve, we took value of t from someinterval [a, b] and plug them into
~r(t) = x(t)~i + y(t)~j + z(t)~k
and the resulting set of vectors will be the position vectors for the pointson the curve.To parameterize a surface, we take points (u, v) on region D in theuv-plane and plug them into
~r(u, v) = x(u, v)~i + y(u, v)~j + z(u, v)~k
The resulting set of vectors will be the position vectors for the points onthe surface S that we are trying to parameterize. This is often called theparameteric representation of the parametric surface S
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parametric surface
The parameteric equation for a surface isx = x(u, v), y = y(u, v), z = (u, v)
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 3 / 20
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Example 1
Identify and sketch the surface with vector equation
~r(u, v) = 2 cos u~i + v~j + 2 sin u~k
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Parametric surface
If a parametric surface S is given by a vector function ~r(u, v), thenthere are two useful families of curves that lie on S , one family with uconstant and the other with v constant.
These families correspond to vertical and horizontal lines in theuv-plane.
If we keep u constant by putting u = u0, then ~r(u0, v) becomes avector function of the single parameter v and defines a curve C1 lyingon S .
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Grid Curves
Similarly, if we keep v constant by putting v = v0, then ~r(u, v0)becomes a vector function of the single parameter u and defines acurve C2 lying on S
We call these ~r(u0, v) and ~r(u, v0) Grid Curves
In fact, when a computer graphs a parametric surface, it usuallydepicts the surface by plotting these grid curves
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Example 2
Give parametric representations for the sphere x2 + y2 + z2 = 30
Solution: Equation of sphere is x2 + y2 + z2 = ⇢2, so ⇢ =p30
Recall the parametric equation of the spherical
x = ⇢ sin� cos ✓
y = ⇢ sin� sin ✓
z = ⇢ cos�
We have the corresponding vector equation
~r(✓,�) =p30 sin� cos ✓~i +
p30 sin� sin ✓~j +
p30 cos�~k
where 0 ✓ 2⇡, 0 � ⇡
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Example 2
~r(✓,�) =p30 sin� cos ✓~i +
p30 sin� sin ✓~j +
p30 cos�~k
The grid curves for a sphere are curves of constant latitude or constantlongitude.
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 8 / 20
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Tangent Plane
Recall the equation of a plane
a(x � x0) + b(y � y0) + c(z � z0) = 0
where the point(x0, y0, z0)
is on the plane and the normal vector of the plane is
~n =< a, b, c >
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 9 / 20
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Tangent Plane
We now find the tangent plane to the parametric surface S at a point P0
with position vector ~r(u0, v0). The surface S is given by a vector function
~r(u, v) = x(u, v)~i + y(u, v)~j + z(u, v)~k
Definition
~ru =@x
@u(u0, v0)~i +
@y
@u(u0, v0)~j +
@z
@u(u0, v0)~k
~rv =@x
@v(u0, v0)~i +
@y
@v(u0, v0)~j +
@z
@v(u0, v0)~k
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 10 / 20
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Example 3
Find the tangent plane to the surface with parameteric equationsx = u2, y = v2, z = u + 2v at the point (1, 1, 3)
Solution:
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 11 / 20
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Example 3
Find the tangent plane to the surface with parameteric equationsx = u2, y = v2, z = u + 2v at the point (1, 1, 3)
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 12 / 20
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Tangent plane-2 (x - l ) - 4cg - l ) -14 (2--3)=0
Surface Area
Now we define the surface area of a general parametric surface. Let’schoose (u⇤i , v
⇤j ) to be the lower left color of small region Rij
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Surface Area
Let ~r⇤u = ~ru(u⇤i , v⇤j ) and ~r⇤v = ~rv (u⇤i , v
⇤j ) be the tangent vectors at point
Pij . The vector of the two edges of the region can be approximated by thevectors �u~r⇤u and �v~r⇤v
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Surface Area
So we approximate Sij by the parallelogram determined by the vectors�u~r⇤u and �v~r⇤v .
The area of this parallelogram is the magnitude of the cross product.
|(�u~r⇤u )⇥ (�v~r⇤v )| = |~r⇤u ⇥ ~r⇤v |�u�v
So the Area of S can be approximated bymX
i=1
nX
j=1
|~r⇤u ⇥ ~r⇤v |�u�v
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 15 / 20
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Surface Area Definition
DefinitionIf a smooth parametric surface S is given by the equation
~r(u, v) = x(u, v)~i + y(u, v)~j + z(u, v)~k , (u, v) 2 D
and S is covered just once as (u, v) ranges throughout the parameterdomain D, then the surface area of S is
A(S) =
ZZ
D|~ru ⇥ ~rv |dA
where ~ru = @x@u~i + @y
@u~j + @z
@u~k and ~rv = @x
@v~i + @y
@v~j + @z
@v~k
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Example 4
Find the surface area of a sphere of radius a.
Solution:
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Example 4
Find the surface area of a sphere of radius a.
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 18 / 20
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Example 4
Find the surface area of a sphere of radius a.
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 19 / 20
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Example 4
Find the surface area of a sphere of radius a.
Xin Li (FSU) Section 16.6 MAC2313 Summer 2020 20 / 20