Surfaces, Part 5

Surfaces and Contour Plots

Part 5: Parametric Surfaces

Just as curves in the plane can be described by parametric representations, so can surfaces in space. For a curve in the plane, we need two functions of a single variable, say, x = x(t) and y = y(t), to describe a one-dimensional set of points parameterized by t ranging over some domain. For a surface in space, we need three functions, each of two variables, say, x = x(u,v), y = y(u,v), and z = z(u,v), with u and v ranging over some two-dimensional domain.

For example, the graph of (x, y, z) = (u - v, u + v, uv) for 0 < u < 2 and 0 < v < 2 is shown in the following figure.

Plot this surface in your worksheet, and rotate it to see it from various angles. Experiment with surfaces parameterized by other functions of u and v.

In Part 3, we described a circular cylinder as being "traced out" by moving a circle in the plane along a line perpendicular to that plane. This gives us a way to parameterize the cylinder -- starting with a one-parameter description of the circle in the xy-plane -- say, x = 4 cos , y = 4 sin -- and then letting z itself be the other parameter: z = z. Since the generating circle can also be described by r = 4, this is very similar to describing the cylinder in cylindrical coordinates.

Use the parameterization just given to plot a cylinder in your worksheet. How does the plot differ in appearance (if at all) from your plot in Part 3?

Choose another cylinder that you plotted in Part 3 -- construct and plot a parametric representation for this cylinder. How does the plot differ in appearance (if at all) from your previous plot of the same surface?

A surface that is the graph of a function of two variables, say, z = f(x,y), can also be thought of parametrically as the set of points (x, y, f(x,y)) for the given domain of x and y. That is, the Cartesian coordinates x and y can be the parameters in a parametric representation.

Plot a parametric representation for the Cobb-Douglas function f(x,y) = over 0 < x < 2, 0 < y < 2. How does the plot differ in appearance (if at all) from your plot in Part 4?

The latitude and longitude coordinates we use for position on the surface of the Earth may be thought of as parameters for a parametric representation of this surface -- which we assume is spherical, for simplicity. These coordinates are similar to spherical coordinates, but with two differences. In spherical coordinates, ranges from 0 to 2, whereas longitude effectively ranges from - to (in radians), with negative values meaning west of Greenwich, England, and positive values meaning east. Similarly, the spherical ranges from 0 at the North Pole to at the South Pole, whereas latitude (in radians) ranges from /2 down to -/2, with positive meaning north and negative south. If let u and v represent latitude and longitude, respectively, both in radians, then we can write u = /2 - and v = - .

Explain why the surface of the Earth (assumed to be a sphere of radius R) can be parameterized by x = - R cos u cos v, y = - R cos u sin v, z = R sin u. [Hint: Use the spherical-to-Cartesian coordinate transformation formulas.] Choose a number for R and plot this parametric surface for appropriate ranges of u and v. How does the appearance of the plot compare to that of the sphere you plotted in Part 2? How does it compare to the illustration in the module page for Part 2?

Describe in words the surface parameterized by x = u cos v, y = u sin v, z = u, 0 < u < 4, 0 < v < 2. Explain how your description relates to the parametric formulas.