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Section 10.5
Parametric Surfaces
Math 21a
February 25, 2008
Announcements
I Problem Sessions: Monday, 8:30 (Sophie); Thursday, 7:30(Jeremy); SC 103b
I Office hours Tuesday, Wednesday 2–4pm SC 323.
I Mathematica assignment due February 29.
Image: Mike Baird
Outline
Explicit versus implicit descriptions
Easy parametrizationsGraphsPlanesOther coordinate surfacesSurfaces of revolution
Other parametrizations
An implicit description of a surface is an equation satisfied by allpoints in the surface.
Example
The unit sphere in R3 is theset of all points (x , y , z) suchthat
x2 + y2 + z2 = 1
Image: dharmesh84
An implicit description of a surface is an equation satisfied by allpoints in the surface.
Example
The unit sphere in R3 is theset of all points (x , y , z) suchthat
x2 + y2 + z2 = 1
Image: dharmesh84
An explicit description of a surface is as the image of a functionr : D → R3, where D is a subset of the plane.
Example
I The unit sphere can be described as the image of two maps:
r+ : D → R3, (x , y) 7→ (x , y ,√
1− x2 − y2)
r− : D → R3, (x , y) 7→ (x , y ,−√
1− x2 − y2)
Here D is the unit disk in the plane:D =
{(x , y)
∣∣ x2 + y2 ≤ 1}
I It can also be described as the image of one map
r : I → R3, (θ, ϕ) 7→ (cos θ sinϕ, sin θ sinϕ, cosϕ)
Here I = [0, 2π]× [0, π].
An explicit description of a surface is as the image of a functionr : D → R3, where D is a subset of the plane.
Example
I The unit sphere can be described as the image of two maps:
r+ : D → R3, (x , y) 7→ (x , y ,√
1− x2 − y2)
r− : D → R3, (x , y) 7→ (x , y ,−√
1− x2 − y2)
Here D is the unit disk in the plane:D =
{(x , y)
∣∣ x2 + y2 ≤ 1}
I It can also be described as the image of one map
r : I → R3, (θ, ϕ) 7→ (cos θ sinϕ, sin θ sinϕ, cosϕ)
Here I = [0, 2π]× [0, π].
An explicit description of a surface is as the image of a functionr : D → R3, where D is a subset of the plane.
Example
I The unit sphere can be described as the image of two maps:
r+ : D → R3, (x , y) 7→ (x , y ,√
1− x2 − y2)
r− : D → R3, (x , y) 7→ (x , y ,−√
1− x2 − y2)
Here D is the unit disk in the plane:D =
{(x , y)
∣∣ x2 + y2 ≤ 1}
I It can also be described as the image of one map
r : I → R3, (θ, ϕ) 7→ (cos θ sinϕ, sin θ sinϕ, cosϕ)
Here I = [0, 2π]× [0, π].
Goals
I Given a surface, find a parametrization r of it
I Given a function r : D → R3, find the image surface.
Outline
Explicit versus implicit descriptions
Easy parametrizationsGraphsPlanesOther coordinate surfacesSurfaces of revolution
Other parametrizations
Parametrizing graphs
If S is the graph of a function f : D → R, then the function can beused for a parametrization:
r : D → R3, (x , y) 7→ (x , y , f (x , y))
The grid lines x = constant and y = constant trace out curves onthe surface.
Advantages/Disadvantages
I Often this is easy
I bad if f is not differentiable at points in D
I sometimes you need more than one
Parametrizing graphs
If S is the graph of a function f : D → R, then the function can beused for a parametrization:
r : D → R3, (x , y) 7→ (x , y , f (x , y))
The grid lines x = constant and y = constant trace out curves onthe surface.
Advantages/Disadvantages
I Often this is easy
I bad if f is not differentiable at points in D
I sometimes you need more than one
Parametrizing graphs
If S is the graph of a function f : D → R, then the function can beused for a parametrization:
r : D → R3, (x , y) 7→ (x , y , f (x , y))
The grid lines x = constant and y = constant trace out curves onthe surface.
Advantages/Disadvantages
I Often this is easy
I bad if f is not differentiable at points in D
I sometimes you need more than one
PlanesAn implicit description of a surface is
n · (r − r0) = 0
A parametric description would be as the image of
r : R2 → R3, (s, t) 7→ r0 + su + tv
Example (Worksheet problem 1)
Write a parameterization for the plane through the point (2,−1, 3)containing the vectors u = 2i + 3j− k and v = i− 4j + 5k.
AnswerTake
r(s, t) = 〈2,−1, 3〉+ s 〈2, 3,−1〉+ t 〈1,−4, 5〉= 〈2 + 2s + t,−1 + 3s − 4t, 3− s + 5t〉
PlanesAn implicit description of a surface is
n · (r − r0) = 0
A parametric description would be as the image of
r : R2 → R3, (s, t) 7→ r0 + su + tv
Example (Worksheet problem 1)
Write a parameterization for the plane through the point (2,−1, 3)containing the vectors u = 2i + 3j− k and v = i− 4j + 5k.
AnswerTake
r(s, t) = 〈2,−1, 3〉+ s 〈2, 3,−1〉+ t 〈1,−4, 5〉= 〈2 + 2s + t,−1 + 3s − 4t, 3− s + 5t〉
PlanesAn implicit description of a surface is
n · (r − r0) = 0
A parametric description would be as the image of
r : R2 → R3, (s, t) 7→ r0 + su + tv
Example (Worksheet problem 1)
Write a parameterization for the plane through the point (2,−1, 3)containing the vectors u = 2i + 3j− k and v = i− 4j + 5k.
AnswerTake
r(s, t) = 〈2,−1, 3〉+ s 〈2, 3,−1〉+ t 〈1,−4, 5〉= 〈2 + 2s + t,−1 + 3s − 4t, 3− s + 5t〉
Example
Find a parametrization for the plane x + y + z = 1.
SolutionThe normal vector is n = 〈1, 1, 1〉; the plane passes through(1, 0, 0). We still need two vectors perpendicular to n: 〈1, 1,−2〉and 〈1,−1, 0〉 will work (there are other choices). We get
r(s, t) = 〈1, 0, 0〉+ s 〈1, 1,−2〉+ t 〈1,−1, 0〉= 〈1 + s + t, s − t,−2s〉
Notice that x(s, t) + y(s, t) + z(s, t) = 1 for all s and t.
Example
Find a parametrization for the plane x + y + z = 1.
SolutionThe normal vector is n = 〈1, 1, 1〉; the plane passes through(1, 0, 0). We still need two vectors perpendicular to n: 〈1, 1,−2〉and 〈1,−1, 0〉 will work (there are other choices). We get
r(s, t) = 〈1, 0, 0〉+ s 〈1, 1,−2〉+ t 〈1,−1, 0〉= 〈1 + s + t, s − t,−2s〉
Notice that x(s, t) + y(s, t) + z(s, t) = 1 for all s and t.
Other coordinate surfaces
The conversion from other coordinate systems to rectangularcoordinates is a kind of parametrization.
Example (Worksheet problem 2)
Write an equation in x , y , and z for the parametric surface
x = 3 sin s y = 3 cos s z = t + 1,
where 0 ≤ s ≤ π and 0 ≤ t ≤ 1.
AnswerThe image is the part of the cylinder x2 + y2 = 9 which also has1 ≤ z ≤ 2 and x ≥ 0.
Other coordinate surfaces
The conversion from other coordinate systems to rectangularcoordinates is a kind of parametrization.
Example (Worksheet problem 2)
Write an equation in x , y , and z for the parametric surface
x = 3 sin s y = 3 cos s z = t + 1,
where 0 ≤ s ≤ π and 0 ≤ t ≤ 1.
AnswerThe image is the part of the cylinder x2 + y2 = 9 which also has1 ≤ z ≤ 2 and x ≥ 0.
Other coordinate surfaces
The conversion from other coordinate systems to rectangularcoordinates is a kind of parametrization.
Example (Worksheet problem 2)
Write an equation in x , y , and z for the parametric surface
x = 3 sin s y = 3 cos s z = t + 1,
where 0 ≤ s ≤ π and 0 ≤ t ≤ 1.
AnswerThe image is the part of the cylinder x2 + y2 = 9 which also has1 ≤ z ≤ 2 and x ≥ 0.
01
23
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2
1.0
1.5
2.0
Surfaces of revolution
These can be parametrized by drawing circles whose radius is thefunction value.
Example
The graph of y = sin x on 0 ≤ x ≤ π is revolved around the x-axis.Find a parametrization of the the surface.
SolutionFor each x0, a circles of radius f (x0) is traced out in the planex = x0. So a parametrization could be
r 7→ [0, π]× [0, 2π]→ R3(x , θ) 7→ (x , f (x) cos θ, f (x) sin θ)
Surfaces of revolution
These can be parametrized by drawing circles whose radius is thefunction value.
Example
The graph of y = sin x on 0 ≤ x ≤ π is revolved around the x-axis.Find a parametrization of the the surface.
SolutionFor each x0, a circles of radius f (x0) is traced out in the planex = x0. So a parametrization could be
r 7→ [0, π]× [0, 2π]→ R3(x , θ) 7→ (x , f (x) cos θ, f (x) sin θ)
Outline
Explicit versus implicit descriptions
Easy parametrizationsGraphsPlanesOther coordinate surfacesSurfaces of revolution
Other parametrizations
Rest of Worksheet problems
Image: Erick Cifuentes