1. Section 17.5 Parameterized Surfaces

2. Recall Parametric curves in 3-space
In rectangular coordinates (x,y,z) we have
We used these to make a helix

3. Parameterized curves in spherical coordinates
In spherical coordinates we have (ρ, , θ)
As parametric equations we let C be the curve
We used these to make a spherical helix

4. Now we are going to move on to parameterized surfaces
What would we get from the following set of parametric equations?
Let’s take a look with Maple

5. In the first two cases we have a parameterized curve in 3 space
In the third case we have a parameterized surface as our parametric equations were in terms of two variables
In rectangular coordinates, parameterized surfaces are given by
where a ≤ u ≤ b and c ≤ v ≤ d
Both u and v are object parameters
Can do in cylindrical and spherical also

6. Let’s see how we can parameterize a sphere
In rectangular coordinates
In spherical coordinates
How about a Torus
One way of thinking
about a Torus is
a circle rotated
around the z-axis
Let’s take a look
in maple

7. Generalized Torus Say we want to create a Torus that is not circular
Essentially we want to rotate some region in a plane around the third axis
(5,1)
(12,6)
(8,14) x z

8. Generalized Torus
We need to make parametric equations for the triangular region
This will be easier to do in cylindrical coordinates
(5,1)
(12,6)
(8,14) x z

9. Parameterizing Planes
The plane through the point with position vector and containing the two nonparallel vectors and is given by
So if
The parametric equations are

10. Parameterizing Surfaces of Revolution
We can create surfaces that have an axis rotational symmetry and circular cross sections to that axis
For example, how about a cone that has a base that is a circle of radius 3 in the xy-plane and a height of 10.
The following structure can be used to revolve a curve around the z axis
This can be modified to revolve around other axes as well

11. Parameter Curves
Parameter curves are obtained by setting one of the parameters to a constant and letting the other vary
Take the following parametric equations
What do they give us?
What would we get if z is held constant?
What would we get if t is held constant?
These parameter curves are cross sections of our parameterized surface