Parametric Representations of Surfaces

Part 1: Parameterizing Surfaces

Surfaces in three dimensional space can be described in many ways -- for example,

• graphs of functions of two variables,
• graphs of equations in three variables, and
• level sets for functions of three variables.

The surface at the right exemplifies all three as

• the graph of the function f(x,y) = x2 - y2,
• the graph of the equation z = x2 - y2, or
• a level set of the function f(x,y,z) = x2 - y2 - z.

On the other hand, some surfaces cannot be represented in any of these ways. The surface at the right, whose technical name is "torus," is an example. In this module, we explore a new -- but familiar -- way of describing surfaces, as parametric mappings of planar regions in space. In particular, we will see that there is a natural way to describe the torus as a parametric surface.

1. Recall that a parameterized curve in the plane is the image of a straight line segment "bent" by some mapping r(t) = (x(t),y(t)). For example, the vector-valued function r(t) = (t,t2) describes a parameterized parabola in the plane. Write down a vector-valued function that describes a circle in the plane. Plot your parametric curve in your worksheet to confirm that it represents a circle.
2. This concept extends to parameterization of a surface in three dimensional space by adding a third coordinate as an output of the vector valued function and a second parameter as an input to the function. Why do you think a second parameter is needed?

Often the two parameters are labeled u and v. The parameterized surface is a vector valued function r(u,v) of two variables, whether written in ijk vector notation or as an ordered triple of functions of u and v. Since each of the variables u and v ranges over an interval, the domain for r(u,v) is a coordinate rectangle, say [a,b] x [c,d], in the uv-plane. (Either or both intervals may be infinitely long.)

1. Every surface that is the graph of a function f(x,y) can also be described parametrically by letting the parameters be x and y. In you r worksheet, confirm that the saddle surface which is the graph of f(x,y) = xy is the same as the graph of r(u,v) = ui + vj + uvk.

Just as we can use Cartesian coordinates as parameters, we can use other coordinate systems as well. The next two steps illustrate this.

1. Use the cylindrical coordinates u =  and v = z to construct a parametric representation of a circular cylinder of radius 2 and height 3. Plot your parametric surface in your worksheet.
2. Use the spherical coordinates and v =  to construct and plot a sphere of radius 2.

Now we consider a parameterization of the torus pictured above before step 1. We can visualize this surface by first imagining a circle of radius a in the xy-plane that runs through the center of the "tube". From each point on this circle, we can reach a circle of points on the surface by making it the center of a circle of radius b, where b < a. This second circle is drawn in a vertical plane that includes the z-axis. In the figure at the right, we show the horizontal circle of radius a in blue and a typical circle of radius b in red. As the red circle travels around the blue one, it sweeps out the entire torus. We take as parameters u and v, respectively, the central angles for the blue and red circles. Then we can construct the parameterization for each point r(u,v) on the surface by adding a vector from the origin to a point on the blue circle and a vector from that point to a point on the corresponding red circle.

1. Explain why the construction just described leads to the parameterization

x(u,v) =  (a + b cos v) cos u,
y(u,v) =  (a + b cos v) sin u,
z(u,v) = b sin v.

What are the domains for u and v? Confirm the parameterization by drawing a torus in your worksheet.

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