This document outlines different ways to parametrize surfaces, including: - Graphs of functions can be parametrized using the function. - Planes can be parametrized using a point on the plane and two vectors. - Coordinate surfaces like cylinders can be parametrized using the coordinate conversions. - Surfaces of revolution can be parametrized using a radius function and angle. The document provides examples of parametrizing planes, cylinders, and surfaces of revolution and outlines other topics like implicit vs explicit descriptions and more complex parametrizations.

1. Section 10.5
Parametric Surfaces
Math 21a
February 25, 2008
Announcements
Problem Sessions: Monday, 8:30 (Sophie); Thursday, 7:30
(Jeremy); SC 103b
Office hours Tuesday, Wednesday 2–4pm SC 323.
Mathematica assignment due February 29.
Image: Mike Baird

2. Outline
Explicit versus implicit descriptions
Easy parametrizations
Graphs
Planes
Other coordinate surfaces
Surfaces of revolution
Other parametrizations

3. An implicit description of a surface is an equation satisfied by all
points in the surface.

4. An implicit description of a surface is an equation satisfied by all
points in the surface.
Example
The unit sphere in R3 is the
set of all points (x, y , z) such
that
x2 + y2 + z2 = 1
Image: dharmesh84

5. An explicit description of a surface is as the image of a function
r : D → R3 , where D is a subset of the plane.

6. An explicit description of a surface is as the image of a function
r : D → R3 , where D is a subset of the plane.
Example
The unit sphere can be described as the image of two maps:
r+ : D → R3 , (x, y ) → (x, y , 1 − x 2 − y 2)
r− : D → R3 , (x, y ) → (x, y , − 1 − x 2 − y 2 )
Here D is the unit disk in the plane:
D = (x, y ) x 2 + y 2 ≤ 1

7. An explicit description of a surface is as the image of a function
r : D → R3 , where D is a subset of the plane.
Example
The unit sphere can be described as the image of two maps:
r+ : D → R3 , (x, y ) → (x, y , 1 − x 2 − y 2)
r− : D → R3 , (x, y ) → (x, y , − 1 − x 2 − y 2 )
Here D is the unit disk in the plane:
D = (x, y ) x 2 + y 2 ≤ 1
It can also be described as the image of one map
r : I → R3 , (θ, ϕ) → (cos θ sin ϕ, sin θ sin ϕ, cos ϕ)
Here I = [0, 2π] × [0, π].

8. Goals
Given a surface, find a parametrization r of it
Given a function r : D → R3 , find the image surface.

9. Outline
Explicit versus implicit descriptions
Easy parametrizations
Graphs
Planes
Other coordinate surfaces
Surfaces of revolution
Other parametrizations

10. Parametrizing graphs
If S is the graph of a function f : D → R, then the function can be
used for a parametrization:
r : D → R3 , (x, y ) → (x, y , f (x, y ))

11. Parametrizing graphs
If S is the graph of a function f : D → R, then the function can be
used for a parametrization:
r : D → R3 , (x, y ) → (x, y , f (x, y ))
The grid lines x = constant and y = constant trace out curves on
the surface.

12. Parametrizing graphs
If S is the graph of a function f : D → R, then the function can be
used for a parametrization:
r : D → R3 , (x, y ) → (x, y , f (x, y ))
The grid lines x = constant and y = constant trace out curves on
the surface.
Advantages/Disadvantages
Often this is easy
bad if f is not differentiable at points in D
sometimes you need more than one

13. Planes
An implicit description of a surface is
n · (r − r0 ) = 0
A parametric description would be as the image of
r : R2 → R3 , (s, t) → r0 + su + tv

14. Planes
An implicit description of a surface is
n · (r − r0 ) = 0
A parametric description would be as the image of
r : R2 → R3 , (s, t) → 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.

15. Planes
An implicit description of a surface is
n · (r − r0 ) = 0
A parametric description would be as the image of
r : R2 → R3 , (s, t) → 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.
Answer
Take
r(s, t) = 2, −1, 3 + s 2, 3, −1 + t 1, −4, 5
= 2 + 2s + t, −1 + 3s − 4t, 3 − s + 5t

16. Example
Find a parametrization for the plane x + y + z = 1.

17. Example
Find a parametrization for the plane x + y + z = 1.
Solution
The 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.

18. Other coordinate surfaces
The conversion from other coordinate systems to rectangular
coordinates is a kind of parametrization.

19. Other coordinate surfaces
The conversion from other coordinate systems to rectangular
coordinates 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.

20. Other coordinate surfaces
The conversion from other coordinate systems to rectangular
coordinates 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.
Answer
The image is the part of the cylinder x 2 + y 2 = 9 which also has
1 ≤ z ≤ 2 and x ≥ 0.

21. 0
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22. Surfaces of revolution
These can be parametrized by drawing circles whose radius is the
function 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.

23. Surfaces of revolution
These can be parametrized by drawing circles whose radius is the
function 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.
Solution
For each x0 , a circles of radius f (x0 ) is traced out in the plane
x = x0 . So a parametrization could be
r → [0, π] × [0, 2π] → R3 (x, θ) → (x, f (x) cos θ, f (x) sin θ)

24. Outline
Explicit versus implicit descriptions
Easy parametrizations
Graphs
Planes
Other coordinate surfaces
Surfaces of revolution
Other parametrizations

25. Rest of Worksheet problems
Image: Erick Cifuentes