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.

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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