How do you parameterize a surface and visualize it in 3-space?

[Nick M]

I'm trying to finish reading/understanding the textbook we used in Calculus III (multivariate), as we only covered chapters 12-18, but I'm stuck on something.

We used McCallum/Hughes-Hallett/Gleason, and I'm referring to section 19.3 (if you have the text) which is about flux integrals over parameterized surfaces. I actually feel strong about my understanding of the flux integral itself using the sum of dot products of the vector field at a location and the area vector generated by a cross product of the partials at that location over the surface, but I need help understanding/visualizing the parameterization of a surface.

I understand how to parameterize a curve in 3-space based on time, but how do you parameterize a surface based on two variables, and visualize this? I can visualize a point drawing a line as t is changed, but the text includes no preface to this section regarding the parameterization of surfaces. Say x, y, and z are dependent upon the two variables s and t. For a line I visualize t as being time, but what would s be when parameterizing a surface?

Does anyone have any tips or articles they could point out?

Thanks.

 

[HallsofIvy]

You can think of s and t as coordinates in the surface. HOW you "parameterize a surface" depends on the problem. There are, in general, an infinite number of ways to parameterize any given surface and which you choose depends both upon how the surface is presented and what you want to use the parameterization for.

If you are given z= f(x,y) then it would be reasonable to use x and y as parameters. A sphere, say $x^2+ y^2+ z^2= R^2$, where you cannot write anyone coordinate as a function of the other two, it would be reasonable to use spherical coordinates: $x= R cos(\theta)sin(\phi)$, $y= R sin(\theta)sin(\phi)$, $z= R cos(\phi)$ with r fixed at R.

 

[Nick M]

You can think of s and t as coordinates in the surface.

Thanks for your response HOI. I think the above quote is what I need help with.

I visualize the parameterization of a curve in 3-space (where the x, y, and z coordinates are a function of t) as a dot that sweeps out the curve as we change the value of t. In effect, I think of a person tossing a ball, and define it's path (the curve) as a function of time.

Is there a visualization technique for how parameters span a surface?

I can visualize the construction of a surface when the parameterization is simple, such as x = s, y = t, z = f(s,t), but how do you approach the visualization when x, y, and z are all dependent upon s and t?

 

[Hurkyl]

I visualize the parameterization of a curve in 3-space (where the x, y, and z coordinates are a function of t) as a dot that sweeps out the curve as we change the value of t. In effect, I think of a person tossing a ball, and define it's path (the curve) as a function of time.
...
but how do you approach the visualization when x, y, and z are all dependent upon s and t?

Why not do exactly the same thing for surfaces as you do for curves?

(Alternatively, you can use a dot to trace a curve, then a curve to trace a surface)

 

[Nick M]

For some reason I'm not visualizing or thinking of it properly.

So taking the example above, I would visualize a standard 3-space cartesian coordinate system where a position corresponds to a point with an x, y, and z value.

Would I be correct by saying we have another space (separate from the above), in this case a 2-space coordinate system in which the two axis' are labeled s and t? A "parameter space"?

Would this be similar to topics in linear algebra related to mapping? We use functions to map points in the parameter space to the x/y/z 3-space?

 

[HallsofIvy]

Or think of an "xy" coordinate system on the z= 0 plane projected onto the z= f(x,y) surface- that will give the equations using x and y themselves as parameters.

 

[Nick M]

But doesn't that only work if x = x, y = y, and z= f(x, y)?

If you have x = g(s,t), y = h(s,t), and z = f(s,t) then z is no longer a function of x and y.

For example, this flux integral...

Find the flux of the vector field F = xi + yj through the surface S, oriented downward and given by

x = 2s, y = s + t, z = 1 + s - t, where 0 ≤ s ≤ 1, 0 ≤ t ≤ 1.

You then have to think of a separate parameter space correct?
I've been trying to still think of the generation of the surface in terms of time as I did with the baseball analogy, and I don't think I can. I suppose I could think of the particle's position as a function of time and say, temperature (two parameters).

I'm feeling better about it, but I definitely can't look at the parameterization of a surface and visualize it as I can the parameterization of a curve.

 

[NoMoreExams]

Why isn't z no longer a function of x and y?

x = 2s, y = s + t, then 1 + x - y = 1 + 2s - s - t = 1 + s - t which is your z?

 

[Nick M]

I think I see what HOI was getting at. Despite how this surface is parameterized, it is a surface in 3-space, and we can define the surface as a function of x and y (either perfectly or approximated as close as we can possibly care about). I guess the trick would be finding the function that generates the surface (in terms of x and y), and also choosing the proper limits of integration that selects the desired portion of the surface.

I feel much more comfortable about the topic now. I was trying to visualize the generation of the surface all as a function of time, but time is only a single parameter and two are required for a surface. I need to include another parameter such as temperature, cost, etc. in my visualization.