A parametric surface is function from a subset of the plane into space.

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.

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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