Surfaces of revolution

Radius in polar coordinates

Dropping back to two-dimensions for a moment, remember polar coordinates (r,$\theta$),

where r is the radius from the origin to the point P, and $\theta$ is the angle from the positive x-axis. You may recall that we can express the radius r in terms of rectangular (Cartesian) coordinates by r = $\sqrt{x^2 + y^2}$. The set of all points where r is constant is a circle of radius r centered around the origin.

Functions that depend only on radius

Let's say we had a function f (x, y) of a special form so that it depended only on the radius r, i.e., depended on x and y only via the expression $\sqrt{x^2 + y^2}$. In this case, if we changed x and y in such a way that r = $\sqrt{x^2 + y^2}$ didn't change, then the value of the function f (x, y) would not change. Combining this observation with our knowledge about r, we conclude that f (x, y) is constant along any circle centered around the origin.

What I've just described is a function of the form f (x, y) = g$\bigl($$\sqrt{x^2 + y^2}$$\bigr)$, where g(r) is some one-variable function. We know, for example, that the function f (x, y) is constant on the circle of radius 2 centered at the origin, because for any point (x, y) where $\sqrt{x^2 + y^2}$ = 2, the value of the function f (x, y) is g(2).

This property makes it simple to graph the surface z = f (x, y) because it follows directly from the graph the curve z = g(r). And you already know how to graph a one-variable function.

To graph f (x, y), we take advantage of the fact that it doesn't change as we rotate around the origin in the xy-plane. In three dimensions, the z-axis would be pointing out of the screen in the above figure. Hence, this rotation corresponds to rotation around the z-axis. The graph of f (x, y) is the graph of g(r) rotated around the z-axis. For this reason, the resulting surface is a called a surface of revolution.

To illustrate, we'll show how the plot of

$${\frac{\sin \sqrt{\smash[b]{x^2+y^2}}}{\sqrt{\smash[b]{x^2+y^2}}+1}}$$

is a surface of revolution. Since f (x, y) depends on x and y only via the combination r = $\sqrt{x^2 + y^2}$, we can rewrite f (x, y) as f (x, y) = g$\bigl($$\sqrt{x^2 + y^2}$$\bigr)$, where

$${\frac{\sin r}{r+1}}$$

Here's a plot of g(r).

In the following demo, you can transform between the plot of g(r) and the plot of f (x, y) by changing the rotation angle $\theta$. (Drag the blue point along the slider.) When $\theta$ = 0, you have the plot of g(r). When $\theta$ = 2$\pi$, you have the plot of f (x, y).

Can you recognize that each of these is a surface of revolution?

• z = $${\frac{1}{1+x^2+y^2}}$$
• z = e^{(x2 + y2)3}
• z = sin($\sqrt{x^2 + y^2}$) - x² - y²